ULNL3

Back to Ultimate Large Numbers List Part II

ULTIMATE LARGE NUMBERS LIST

PART III

Cascading-E, Extended Cascading-E & Beyond

finite numbers of order type w^w to any number up to and including the absolutely infinite ...

E100#^#100

E100### ... ###100 w/100 #s

godgahlah

                This number pushes the limit of xE#, and soon we will be beyond what we can practically express even in xE#. However this is just the beginning of Cascading-E Notation. In Cascading-E we can write a godgahlah more simply as E100#^#100. This number can be approximated in Array Notation as <10,100,99,99,...,99> w/99 99s. This makes it an array with 101 entries. It is thus larger than goobol = <10,100(1)2>. On googology wiki this number is approximated as <100,101(1)2> which is <100,100,...,100> w/101 100s.

E100,000#^#100,000

E100,000#### ... ... ####100,000 w/100,000 #s

godgahlahgong

            Also can be written in cascading-E as E100,000#^#100,000.

{10, {10,10(1)2} (1)2}

iteralplex

The plex version of an iteral. This is one of the 120 original named numbers by Jonathan Bowers.

E100#^#100#2

E100######## ... ... ... ... ... ... ... ... ... ... ... ... ########100 

w/godgahlah #s

grand godgahlah

            Also can be written E100#^#100#2 = E100#^#(E100#^#100) = E100#^#godgahlah. This is usually the earliest use of "grand" to represent a single recursive step, as creating a new root for every kind of recursion quickly exhausts roots. From here on in if we have @100#n = "googolism", then "grand googolism" will be @100#(n+1).

E100,000#^#100,000#2

E100,000######## ... ... ... ... ... ... ... ... ... ... ... ... ########100,000 w/godgahlahgong #s

grand godgahlahgong

            Alternatively, E100,000#^#100,000#2.

E100#^#100#3

E100#grand godgahlah100

grand grand godgahlah

            This is E100#^#100#3 in Cascading-E. We can also called this number two-ex-grand godgahlah, although this doesn't really shorten the name.

E100,000#^#100,000#3

E100,000#grand godgahlahgong100,000

grand grand godgahlahgong

                This is E100,000#^#100,000#3. This was the highest official entry with the original release of my Ultimate Large Numbers List. That being said someone out there probably suggested ...

1+E100,000#^#100,000#3

grand grand godgahlahgong and one

                In an earlier version of this list, this number was mentioned unofficially (non-entry) as an example of the first really large number on the way to infinity. Adding one to googolism's is a common knee-jerk response from non-googologist's. By mentioning such a number, non-googologist's imply that "there is no largest number" and therefore there is no point in trying to make one. By claiming that they have "named the largest number by adding one" they hope to lampoon the endeavor entirely.

                What they fail to realize is that googologist's are not trying to make the "largest number", but rather simply enjoy making extremely large numbers, and competing with fellow googologist's in this effort. Secondly, there is a big difference from adding one to someone else's googolism and building your own from the ground up. This is equivalent to "standing on the shoulders of giants" in the worst possible sense. There is generally a "gentlemen's agreement" in googology which says you should not make reference to the other person's number in your definition in an attempt to "trump" them, since any number can be trivially made larger by adding one to it (including some infinities I might add). The second rule is you don't get bragging rights if your number is constructable with methods already established by a competitor. In a sense, adding one contributes nothing to the discussion of large numbers. The idea isn't just to make the largest number, but also to keep devising new and better ways to generate large numbers. So it is a misconception that googologist's are only trying to find the largest number. Rather googologist's are always trying to find a larger number, and not just a slightly larger number, but a colossally mind-bogglingly crushing SUPER ULTRA MEGA NE PLUS ULTRA Larger Number ... and that just for a warm up... 

2+E100,000#^#100,000#3

grand grand godgahlahgong and two

            ...because for every googolism, there will be a +1er to add one to it. See (grand grand godgahlahgong and one). Moving on...

2E100,000#^#100,000#3

two grand grand godgahlahgong

            This is a step up from the +1ers, as we are now adding a grand grand godgahlahgong to a grand grand godgahlahgong. We can truncate this as two times a grand grand godgahlahgong. Since our ordinary number sense is logarithmic however, this isn't really an impressive improvement. If the grand grand godgahlahgong is represented by a solid packed sphere of fundamental particles, two grand grand godgahlahgong, would be like having two of these spheres. Does it really seem that much larger, no matter how large the original number was?

3E100,000#^#100,000#3

three grand grand godgahlahgong

            How about 3 times a grand grand godgahlahgong...

7E100,000#^#100,000#3

seven grand grand godgahlahgong

            How about 7 times a grand grand godgahlahgong...

(E100,000#^#100,000#3)^2

grand grand godgahlahgong squared

            This is the next stage in the naive non-googologist's attempt to extend a target googolism. Basically an attempt to pre-empt the impulse to say "two times your number" and jump to "your number times your number". Now you can imagine our sphere of particles compare to a grand grand godgahlahgong of such spheres!!! o_0;

            This will tend to get a visceral reaction, especially since we can't even imagine how big this number is. Despite that fact however, this is just squaring, so any grade school kid could come up with that. Furthermore try to remember that if we began by looking at a single fundamental particle and zoomed out until we reached a grand grand godgahlahgong particles, we'd only have to zoom out twice as long to reach a grand grand godgahlahgong squared. Given how mind-crushingly long the zoom out would be, the fact that it would take twice as long barely makes a difference. Hence, from a googological point of view there is virtually no difference between these numbers. Moving on...

(E100,000#^#100,000#3)^3

grand grand godgahlahgong cubed

            What's better than a square ... a cube...

(E100,000#^#100,000#3)^4

grand grand godgahlahgong to the fourth

            or a 4th power ...

(E100,000#^#100,000#3)^(E100,000#^#100,000#3)

grand grand godgahlahgong to the power of itself

            Next up along the hierarchy of naive responses is raising the number to its own power. Again, with numbers this large it has virtually no effect. That probably sounds mind boggling but just imagine N, then N^N, then (N^N)^(N^N), and continue in this manner for N steps, and call this number M. Someone comes along and says "I could trounce M, how about M^M". Are you really going to be impressed by their response? Ditto for googologist's who've gone much much further ...

(E100,000#^#100,000#3)^^(E100,000#^#100,000#3)

power tower of grand grand godgahlahgong's a grand grand godgahlah terms high

            This one is a classic. Even non-googologist's eventually realize that simply raising a number to the power of itself isn't really that competitive for certain large numbers, especially after they have at least been exposed toGraham's Number. So next they decide to create a power tower of this number, naively thinking this will trounce the competition. But for anything above Graham's Number this already has virtually no effect. If "N" is any googolism, you can bet someone out there will mention N^^N as a response. Rule 34 of googolism states: "If a googolism exists, there is a salad number for it, ... no exceptions". Moving on...

(E100,000#^#100,000#3)^^^^^ ... ^^^^^^(E100,000#^#100,000#3)

w/(E100,000#^#100,000#3) ^s

            Time to bring out the big guns. Our intrepid non-googologist has now gotten wind of Knuth-up arrow notation. They promptly take N and generate N^^^...^^^N w/N ^s. As any googologist worth his/her salt knows, no matter how amazingly powerful a "method" is in ordinary terms, it will always be obsolete moments later. The general rule of thumb is, never use a method that occurred earlier in the game to extend a number late in the game. It's effect will always be negligible relative to the current methods. Even the next entry is vastly larger despite making only a minute change...

E100,000#^#(1+E100,000#^#100,000#2)

            So the non-googologist actually realizes that they can't use any methods other than the most recently established one. Great. Now they attempt to apply that principle and ... miss the point. Since a grand grand godgahlahgong is E100,000####...####100,000 with a grand godgahlahgong #s, a naive response might be to have E100,000####...#####100,000 with a grand godgahlahgong +1 #s. This isn't much of an improvement over just adding one to the number! This response however is still mind-bogglingly better than the last attempt.

 To prove this just consider the following. Let n=grand godgahlahgong, and N=grand grand godgahlahgong.By definition N = E100,000#^(n)100,000. We first observe that:

E100,000#^(n+1)100,000 > E100,000#^(n+1)3 = E100,000#^(n)100,000#^(n)100,000 

> E100,000#^(n)100,000#100,000 > E100,000#^(n)100,000#2 

= E100,000#^(n)(E100,000#^(n)100,000) = E100,000#^(n)N

                Next we work from the other end. We can say that:

<N,N,N> < E(N)N##N < E100,000##(N+1) = E100,000##100,000#N <

E100,000###N < E100,000#^(n)N 

So we conclude...

<grand grand godgahlahgong,3(1)2> < E100,000#^(grand godgahlahgong+1)100,000

E100,000#^#100,000#4

grand grand grand godgahlahgong

            This is the obvious response to a grand grand godgahlahgong. This number is mind-bogglingly more massive. However, just like all the previous attempts to extend a grand grand godgahlahgong this is still too tame an extension, because it employs a method we have been using since the beginning: basic recursion. At this point, the recursion is so complex that applying one additional recursion doesn't make much of a difference from a googologist's point of view. Okay so how about...

E100,000#^#100,000#5

grand grand grand grand godgahlahgong

                Same problem as before. Still just basic recursion... how about ...

E100#^#100#100

grandgahlah

                This is the smallest new number defined with Cascading-E Notation. A grandgahlah may also be called a grand grand grand ... grand grand grand godgahlah where there are 99 grands. For this we can use the short hand name ninety-nine-ex-grand godgahlah. This still isn't a significant improvement from agodgahlah since we already introduced proto-hyperions a long time ago.

E100#^#100#101

hundred-ex-grand godgahlah

                Trivial, but it just sounds cool...

E100,000#^#100,000#11,863

            grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand 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grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand 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grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand 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grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand godgahlahgong 

            Also known as eleven-thousand-eight-hundred-sixty-two-ex-grand godgahlahgong. Just kidding. This is a number based on the original trail of grands I had left at the end of my ULNL with it's initial release to suggest the next direction of development. Obviously, listing out the grands is inefficient. Still it's mind-boggling to think how all these "grand"s modify "godgahlahgong". Even so, this kind of stuff is easily trumped by the many numbers that follow as we begin to explore the consequences of Cascading-E notation... 

E100#^#100#(1+E100#^#100)

godgahlah-ex-grand godgahlah

            This response is far far better than any of the previous responses... but it's still a naive extension. Why? Because it still only requires a single extra proto-hyperion to achieve these results, and with two proto-hyperions we get even further.

E100#^#100#(1+E100#^#100#2)

grand-godgahlah-ex-grand godgahlah

For a naive continuation, let's start nesting this -ex-grand construction and see how far that gets us ...

E100#^#100#(1+E100#^#100#3)

grand-grand-godgahlah-ex-grand godgahlah

This is also two-ex-grand-godgahlah-ex-grand godgahlah ... let's see where that leads ...

E100#^#100#(1+E100#^#100#4)

grand-grand-grand-godgahlah-ex-grand godgahlah

Also three-ex-grand-godgahlah-ex-grand godgahlah.

E100#^#100#(1+E100#^#100#5)

grand-grand-grand-grand-godgahlah-ex-grand godgahlah

Also four-ex-grand-godgahlah-ex-grand godgahlah. We are basically still in the ballpark of godgahlah and barely moved ...

E100#^#100#100#2

grand grandgahlah

                E100#^#100#100#2 = E100#^#100#(E100#^#100#100) = E100#^#100#(grandgahlah). The great grandgahlah is the grandgahlahth member of the godgahlah series.

E100#^#100#100#3

grand grand grandgahlah

Grand's can be stacked just as before. Let's skip on ahead to ...

E100#^#100#100#100

greagahlah

The pattern here should be clear. This is greagol + godgahlah to stand for E100#^# followed by 100#100#100. Adapting more of these is simple. Drop "gol" and "god" and simply concatenate the remaining letters. Let's see what this leads to ...

E100#^#100#100#100#2

grand greagahlah

As usual we can have a "grand" version of greagahlah as well ...

E100#^#100#100#100#3

grand grand greagahlah

and a "grand grand" version. Moving on ...

E100#^#100#100#100#100

gigangahlah

The next major construction. gigan(gol) + (god)gahlah = gigangahlah. 

E100#^#100#100#100#100#2

grand gigangahlah

another "grand" version. From here on out you can assume for each of these there is at least one grand version implicitly.

E100#^#100#100#100#100#100

gorgegahlah

gorge(gol) + (god)gahlah = gorgegahlah.

E100#^#100#100#100#100#100#100

gulgahlah

gul(gol) + (god)gahlah = gulgahlah.

E100#^#100#100#100#100#100#100#100

gaspgahlah

gasp(gol) + (god)gahlah = gaspgahlah.

E100#^#100#100#100#100#100#100#100#100

ginorgahlah

ginor(gol) + (god)gahlah = ginorgahlah.

E100#^#100##100

gugoldgahlah

A slight breach of form here, since we need "gold" to indicate what row we are on. From here on out we can simply use "googolism" + "gahlah" to continue. The continuations from here should make sense ...

E100#^#100##100#100

graatagoldgahlah

graatagold + godgahlah. Just take graatagold and replace the 'E' with 'E100#^#'.

E100#^#100##100#100#100

greegoldgahlah

greegold + godgahlah. 'E100#^#' + '100##100#100#100'.

E100#^#100##100#100#100#100

grinningoldgahlah

grinningold + godgahlah. 'E100#^#' + '100##100#100#100#100'.

E100#^#100##100#100#100#100#100

golaagoldgahlah

golaagold + godgahlah. 'E100#^#' + '100##100#100#100#100#100'.

E100#^#100##100#100#100#100#100#100

gruelohgoldgahlah

gruelohgold + godgahlah. 'E100#^#' + '100##100#100#100#100#100#100'.

E100#^#100##100#100#100#100#100#100#100

gaspgoldgahlah

gaspgold + godgahlah. 'E100#^#' + '100##100#100#100#100#100#100#100'.

E100#^#100##100#100#100#100#100#100#100#100

ginorgoldgahlah

ginorgold + godgahlah. 'E100#^#' + '100##100#100#100#100#100#100#100#100'.

E100#^#100##100##100

gugolthragahlah

gugolthra + godgahlah. 'E100#^#' + '100##100#100'. We could go through all the "thra's", but let's go a little faster than that. We still are only just crawling out of the shadow of godgahlah as far as Cascading-E Notation is concerned ...

E100#^#100##100##100##100

gugolteslagahlah

gugoltesla + godgahlah. 'E100#^#' + '100##100##100##100'.

E100#^#100##100##100##100##100

gugolpetagahlah

gugolpeta + godgahlah. 'E100#^#' + '100##100##100##100##100'.

E100#^#100##100##100##100##100##100

gugolhexagahlah

gugolhexa + godgahlah. 'E100#^#' + '100##100##100##100##100##100'.

E100#^#100##100##100##100##100##100##100

gugolheptagahlah

gugolhepta + godgahlah. 'E100#^#' + '100##100##100##100##100##100##100'.

E100#^#100##100##100##100##100##100##100##100

gugoloctagahlah

gugolocta + godgahlah. 'E100#^#' + '100##100##100##100##100##100##100##100'.

E100#^#100###100

throogahlah

throogol + godgahlah. 'E100#^#' + '100###100'. Here we've gone back to dropping "gol" at the end of the name. Let's take this to it's logical limit ...

E100#^#100####100

teroogahlah

teroogol + godgahlah. 'E100#^#' + '100####100'.

E100#^#100#####100

petoogahlah

petoogol + godgahlah. 'E100#^#' + '100#####100'.

E100#^#100######100

ectoogahlah

ectoogol + godgahlah. 'E100#^#' + '100######100'.

E100#^#100#######100

zettoogahlah

zettoogol + godgahlah. 'E100#^#' + '100#######100'.

E100#^#100########100

yottoogahlah

yottoogol + godgahlah. 'E100#^#' + '100########100'.

E100#^#100#^#100

godgahlahgahlah / gotrigahlah

godgahlah + godgahlah = godgahlahgahlah. 'E100#^#' + '100#^#100' = 'E100#^#100#^#100'. Obviously we've reached the limit of this approach. So let's begin a new strategy. We can use multipliers to repeat a delimiter. Inserting "tri" in the middle will give us 3 arguments separated by #^#'s. So gotrigahlah = E100#^#100#^#100. We've finally stacked #^# recursion on top of #^# recursion. But we are still just getting started with Cascading-E Notation (E^). We haven't even introduced hyperion multiplication yet ... 

E100#^#100#^#100#2

grand gotrigahlah

The obligatory "grand" version just to remind you we can still do this ...

E100#^#100#^#100#3

grand grand gotrigahlah

grand's can be arbitrarily stacked on basically any googolism in ExE including ones that already have grands ...

E100#^#100#^#100#100

grantrigahlah

grangol + gotrigahlah. Here we go again ...

E100#^#100#^#100#100#2

great grantrigahlah

might as well. moving on ...

E100#^#100#^#100#100#100

greatrigahlah

greagol + gotrigahlah. 'E100#^#100#^#' + '100#100#100'.

E100#^#100#^#100#100#100#100

gigantrigahlah

gigangol + gotrigahlah. 'E100#^#100#^#' + '100#100#100#100'.

E100#^#100#^#100##5

gorgetrigahlah

gorgegol + gotrigahlah. 'E100#^#100#^#' + '100##5'.

E100#^#100#^#100##6

gultrigahlah

gulgol + gotrigahlah. 'E100#^#100#^#' + '100##6'.

E100#^#100#^#100##7

gasptrigahlah

gaspgol + gotrigahlah. 'E100#^#100#^#' + '100##7'.

E100#^#100#^#100##8

ginortrigahlah

ginorgol + gotrigahlah. 'E100#^#100#^#' + '100##8'.

E100#^#100#^#100##100

gugoldtrigahlah

gugold + gotrigahlah. 'E100#^#100#^#' + '100##100'. The pattern should be clear, so let's speed things up a bit ...

E100#^#100#^#100###100

throotrigahlah

throogol + gotrigahlah. 'E100#^#100#^#' + '100###100'.

E100#^#100#^#100####100

terootrigahlah

teroogol + gotrigahlah. 'E100#^#100#^#' + '100####100'.

E100#^#100#^#100#####100

petootrigahlah

petoogol + gotrigahlah. 'E100#^#100#^#' + '100#####100'.

E100#^#100#^#100######100

ectootrigahlah

ectoogol + gotrigahlah. 'E100#^#100#^#' + '100######100'.

E100#^#100#^#100#######100

zettootrigahlah

zettoogol + gotrigahlah. 'E100#^#100#^#' + '100#######100'.

E100#^#100#^#100########100

yottootrigahlah

yottoogol + gotrigahlah. 'E100#^#100#^#' + '100########100'.

E100#^#100#^#100#^#100

gotergahlah

godgahlah + gotrigahlah = gotergahlah.

E100#^#100#^#100#^#100#2

grand gotergahlah

grand + gotergahlah = 'E100#^#100#^#100#^#100' + '#2'.

E100#^#100#^#100#^#100#100

grantergahlah

grangol + gotergahlah.

E100#^#100#^#100#^#100#100#2

grand grantergahlah

grand + grantergahlah.

E100#^#100#^#100#^#100#100#100

greatergahlah

greagol + gotergahlah.

E100#^#100#^#100#^#100#100#100#100

gigantergahlah

gigangol + gotergahlah.

E100#^#100#^#100#^#100##5

gorgetergahlah

gorgegol + gotergahlah.

E100#^#100#^#100#^#100##6

gultergahlah

gulgol + gotergahlah.

E100#^#100#^#100#^#100##7

gasptergahlah

gaspgol + gotergahlah.

E100#^#100#^#100#^#100##8

ginortergahlah

ginorgol + gotergahlah.

E100#^#100#^#100#^#100##100

gugoldtergahlah

gugold + gotergahlah.

E100#^#100#^#100#^#100###100

throotergahlah

throogol + gotergahlah.

E100#^#100#^#100#^#100####100

terootergahlah

teroogol + gotergahlah.

E100#^#100#^#100#^#100#####100

petootergahlah

petoogol + gotergahlah.

E100#^#100#^#100#^#100######100

ectootergahlah

ectoogol + gotergahlah.

E100#^#100#^#100#^#100#######100

zettootergahlah

zettoogol + gotergahlah.

E100#^#100#^#100#^#100########100

yottootergahlah

yottoogol + gotergahlah.

E100#^#100#^#100#^#100#^#100

gopeggahlah

Let's speed along now that we get the basic pattern. We can increase the multiplier to quickly stack #^# level recursion. we will use peg, hex, hep, aht, enn, and dek. god + peg + gahlah = gopeggahlah.

E100#^#100#^#100#^#100#^#100#^#100

gohexgahlah

god + hex + gahlah = gohexgahlah.

E100#^#100#^#100#^#100#^#100#^#100#^#100

gohepgahlah

god + hep + gahlah = gohepgahlah.

E100#^#100#^#100#^#100#^#100#^#100#^#100#^#100

go-ahtgahlah

god + aht + gahlah = go-ahtgahlah.

E100#^#100#^#100#^#100#^#100#^#100#^#100#^#100#^#100

go-enngahlah

god + enn + gahlah = go-enngahlah.

{10,10(1)10}

emperal

The emperal was one of the original 120 named numbers by Jonathan Bowers. It falls slightly below a godekahlah. An emperal may be expanded to: 

{10,10,10,10,10,10,10,10,10,10(1)9}

<10,100(1)10>

This is a proposed lowerbound for godekahlah in BEAF. If this bound is true, then it is trivially larger than an emperal. This number expands to <10,10,...,10(1)9> w/100 10s.

E100#^#100#^#100#^#100#^#100#^#100#^#100#^#100#^#100#^#100

godekahlah

god + dek + gahlah = godekahlah.

E100#^#*#100

godgoldgahlah

A godgoldgahlah is approximately equal to <10,100(1)100>.

{10,10 (1) {10,10 (1) 10}}

emperalplex

The plex version of an emperal. This would be approximately E100#^#*#emperal in Cascading-E Notation. Since an emperal is itself approximately E100#^#*#10 this means this number can be approximated as E100#^#*#10#2. Clearly this is smaller than 100#^#*#100#2, which is vanishingly small compare to the next entry ...

E100#^#*#100#^#*#100

gotrigoldgahlah / tristo-godgoldgahlah

Approximately equal <10,100(1)100,2>. 

E100#^#*#100#^#*#100#^#*#100

gotergoldgahlah / teristo-godgoldgahlah

Approximately equal <10,100(1)100,3>.

E100#^#*##5

gopeggoldgahlah / pesto-godgoldgahlah

Approximately equal <10,100(1)100,4>.

E100#^#*##6

gohexgoldgahlah / existo-godgoldgahlah

Approximately equal <10,100(1)100,5>.

E100#^#*##7

gohepgoldgahlah / episto-godgoldgahlah

Approximately equal <10,100(1)100,6>.

E100#^#*##8

go-ahtgoldgahlah / ogisto-godgoldgahlah

Approximately equal <10,100(1)100,7>.

E100#^#*##9

godenngoldgahlah / ennisto-godgoldgahlah

Approximately equal <10,100(1)100,8>.

E100#^#*##10

godekagoldgahlah / dekisto-godgoldgahlah

Approximately equal <10,100(1)100,9>.

{10,10(1)10,10}

hyperal

One of the original Bowerisms. It is believed to fall between E100#^#*##10 and E100#^#*##11.

E100#^#*##100

godthroogahlah

Approximately equal <10,100(1)100,99>.

{ 10 , 10 (1) 10 , { 10 , 10 (1) 10 , 10 } }

hyperalplex

This is bigger than a godthroogahlah but less than a grand godthroogahlah.

E100#^#*###100

godteroogahlah

Approximately equal <10,100(1)100,99,99>. Not to be confused with gotergahlah = E100#^#*#4.

E100#^#*####100

godpetoogahlah

Approximately equal <10,100(1)100,99,99,99>.

E100#^#*#####100

go-ectoogahlah

Approximately equal <10,100(1)100,99,99,99,99>.

E100#^#*######100

godzettoogahlah

Approximately equal <10,100(1)100,99,99,99,99>.

E100#^#*#######100

godyottoogahlah

Approximately equal <10,100(1)100,99,99,99,99,99>.

E100#^#*#^#100

deutero-godgahlah

This is the first number to introduce the "deutero" operator. This duplicates the hypernion, leading to two copies multiplied together. This is believed to be approximately equal to <10,100(1)(1)2>. That is a two-row array equal to <10,10,...,10(1)10,10,...,10> w/100 10s per row. This makes this number much much larger than a hyperal. This number is still much much smaller than a dutridecal.

E100#^#*#^#*#100

deutero-godgoldgahlah

Approximately equal to <10,100(1)(1)100>.

E100#^#*#^#*##100

deutero-godthroogahlah

Approximately equal to <10,100(1)(1)100,99>.

{3,3,3(1)3,3,3(1)3,3,3}

dutritri

One of Bowers' originals.

{10,10,10(1)10,10,10(1)10,10,10}

dutridecal

One of Bowers' originals.

E100#^#*#^#*###100

deutero-godteroogahlah

Approximately equal to <10,100(1)(1)100,99,99>.

E100#^#*#^#*####100

deutero-godpetoogahlah

Approximately equal to <10,100(1)(1)100,99,99,99>.

E100#^#*#^#*#####100

deutero-go-ectoogahlah

Approximately equal to <10,100(1)(1)100,99,99,99,99>.

E100#^#*#^#*######100

deutero-gozettoogahlah

Approximately equal to <10,100(1)(1)100,99,99,99,99,99>.

E100#^#*#^#*#######100

deutero-goyottoogahlah

Approximately equal to <10,100(1)(1)100,99,99,99,99,99,99>.

E100#^#*#^#*#^#100

trito-godgahlah

Approximately equal to <10,100(1)(1)(1)2>. Thats <100&10 (1) 100&10 (1) 100&10 > or three rows of a hundred 10s. This number exceeds a dutridecal. Now that we have established how to create the intermediate cases, let's skip to the "power multipliers". deutero is equivalent to squaring the hypernion, trito is equivalent to cubing the hypernion, let's see where that leads ...

E100#^#*#^#*#^#*#^#100

teterto-godgahlah

Approximately equal to <10,100(1)(1)(1)(1)2>.

E100#^#*#^#*#^#*#^#*#^#100

pepto-godgahlah

Approximately equal to <10,100(1)(1)(1)(1)(1)2>.

E100#^#*#^#*#^#*#^#*#^#*#^#100

exto-godgahlah

Approximately equal to <10,100(1)(1)(1)(1)(1)(1)2>.

E100#^##7

epto-godgahlah

Approximately equal to <10,100(1)(1)(1)(1)(1)(1)(1)2>.

E100#^##8

ogdo-godgahlah

Approximately equal to <10,100(1)(1)(1)(1)(1)(1)(1)(1)2>.

E100#^##9

ento-godgahlah

Approximately equal to <10,100(1)(1)(1)(1)(1)(1)(1)(1)(1)2>.

{10,10(2)2}

xappol

One of Bowers original 120 googolisms. This is a 10x10 & 10s. It is believed to be slightly smaller than a dekato-godgahlah, which is believed to be larger since it is predicted to have 100 entry rows instead of 10 entry rows. It would be a 10x100 & 10s approximately.

E100#^##10

dekato-godgahlah

Approximately equal to <10,100(1)(1)(1)(1)(1)(1)(1)(1)(1)(1)2>.

E100#^##20

isosto-godgahlah

Here is the 20x power multiplier. Let's move on to the next major milestone.

E100#^##100

gridgahlah / hecato-godgahlah

Approximately equal to <10,100(2)2>. This approximation is a 100 x 100 & 10s. This makes it larger than a xappol.

{10,{10,10(2)2}(2)2}

xappolplex

A xappol x xappol & 10s. This would be approximately equal to E100#^##xappol, making it larger than a gridgahlah but much much smaller than a gridtrigahlah.

E100#^##100#^##100

gridtrigahlah

Approximately equal to <10,100(2)3>.

E100#^##100#^##100#^##100

gridtergahlah

Approximately equal to <10,100(2)4>.

E100#^##*#5

gridpeggahlah

Approximately equal to <10,100(2)5>.

E100#^##*#6

gridhexgahlah

Approximately equal to <10,100(2)6>.

E100#^##*#7

gridhepgahlah

Approximately equal to <10,100(2)7>.

E100#^##*#8

grid-ahtgahlah

Approximately equal to <10,100(2)8>.

E100#^##*#9

grid-enngahlah

Approximately equal to <10,100(2)9>.

E100#^##*#10

grid-dekgahlah

Approximately equal to <10,100(2)10>.

E100#^##*#^##100

deutero-gridgahlah

Approximately equal to <10,100(2)(2)2>. Approximately equal to 2 planes of 100x100 10s in array notation.

{3,3(3)2}

{3,3,3(1)3,3,3(1)3,3,3(2)3,3,3(1)3,3,3(1)3,3,3(2)3,3,3(1)3,3,3(1)3,3,3}

dimentri

This expands to {3,3(2)(2)(2)2} which becomes 3 planes. This is much much larger than the two planes of deutero-gridgahlah but it's much smaller than the 3 planes of a trito-gridgahlah. This can be imagined as a 3x3x3 cube, like a Rubik's Cube, where each of the 27 cubes contains a 3.

E100#^##*#^##*#^##100

trito-gridgahlah

Approximately equal to <10,100(2)(2)(2)2>.

E100#^##*#^##*#^##*#^##100

teterto-gridgahlah

Approximately equal to <10,100(2)(2)(2)(2)2>.

E100#^###5

pepto-gridgahlah

Approximately equal to <10,100(2)(2)(2)(2)(2)2>.

E100#^###6

exto-gridgahlah

Approximately equal to <10,100(2)(2)(2)(2)(2)(2)2>.

E100#^###7

epto-gridgahlah

Approximately equal to <10,100(2)(2)(2)(2)(2)(2)(2)2>.

E100#^###8

ogdo-gridgahlah

Approximately equal to <10,100(2)(2)(2)(2)(2)(2)(2)(2)2>.

E100#^###9

ento-gridgahlah

Approximately equal to <10,100(2)(2)(2)(2)(2)(2)(2)(2)(2)2>.

{10,10(3)2} = 10^3 & 10s

colossal

A colossal is a 3-dimensional array of 10s of edge length 10. It has 10^3 or 1000 entries. That's a lot, though that is manageable and could be theoretically written out in full ... though it would be pretty tedious. This is just slightly smaller than dekato-gridgahlah which has 10 planes, but they are of size 100x100 instead of 10x10.

E100#^###10

dekato-gridgahlah

Approximately equal to <10,100(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)2>.

E100#^###20

isosto-gridgahlah

Approximately equal to <10,100(2)...(2)2> w/20 (2)s.

E100#^###100

kubikahlah / hecato-gridgahlah

Approximately equal to <10,100(3)2>. This would be a 100 x 100 x 100 & 10s. This would be much much larger than a dimentri, since a dimentri only has 3 planes, whereas this has 100.

{10,{10,10(3)2}(3)2} = colossal^3 & 10s

colossalplex

A 3-dimensional cube of edge length colossal all of 10s. This would fall below E100#^###100#2, which would be approximately a 3-dimensional cube of edge length kubikahlah all of 10s.

E100#^###100#^###100

kubitrigahlah

Approximately equal to <10,100(3)3>.

E100#^###100#^###100#^###100

kubitergahlah

Approximately equal to <10,100(3)4>.

E100#^###*#5

kubipeggahlah

Approximately equal to <10,100(3)5>.

E100#^###*#6

kubihexgahlah

Approximately equal to <10,100(3)6>.

E100#^###*#7

kubihepgahlah

Approximately equal to <10,100(3)7>.

E100#^###*#8

kubi-ahtgahlah

Approximately equal to <10,100(3)8>.

E100#^###*#9

kubi-enngahlah

Approximately equal to <10,100(3)9>.

E100#^###*#10

kubi-dekahlah

Approximately equal to <10,100(3)10>.

E100#^###*#^###100

deutero-kubikahlah

Approximately equal to <10,100(3)(3)2>. This would be a "double cube" of 100x100x100. This is the beginnings of extending into the 4th dimension.

E100#^###*#^###*#^###100

trito-kubikahlah

Approximately equal to <10,100(3)(3)(3)2>.

E100#^###*#^###*#^###*#^###100

teterto-kubikahlah

Approximately equal to <10,100(3)(3)(3)(3)2>.

E100#^####5

pepto-kubikahlah

Approximately equal to <10,100(3)(3)(3)(3)(3)2>.

E100#^####6

exto-kubikahlah

Approximately equal to <10,100(3)(3)(3)(3)(3)(3)2>.

E100#^####7

epto-kubikahlah

Approximately equal to <10,100(3)(3)(3)(3)(3)(3)(3)2>.

E100#^####8

ogdo-kubikahlah

Approximately equal to <10,100(3)(3)(3)(3)(3)(3)(3)(3)2>.

E100#^####9

ento-kubikahlah

Approximately equal to <10,100(3)(3)(3)(3)(3)(3)(3)(3)(3)2>.

E100#^####10

dekato-kubikahlah

Approximately equal to <10,100(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)2>.

E100#^####20

isosto-kubikahlah

Approximately equal to <10,100(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)2>.

E100#^####100

quarticahlah / hecato-kubikahlah

Approximately equal to <10,100(4)2>. Enter the 4th dimension. This would be equivalent to a 100x100x100x100 tesseract in Array Notation.

E100#^####100#^####100

quartitrigahlah

Approximately equal to <10,100(4)3>. #^#### level recursion stacked on top of #^#### level recursion.

E100#^####100#^####100#^####100

quartitergahlah

Approximately equal to <10,100(4)4>.

E100#^####*#5

quartipeggahlah

Approximately equal to <10,100(4)5>.

E100#^####*#6

quartihexgahlah

Approximately equal to <10,100(4)6>.

E100#^####*#7

quartihepgahlah

Approximately equal to <10,100(4)7>.

E100#^####*#8

quarti-ahtgahlah

Approximately equal to <10,100(4)8>.

E100#^####*#9

quarti-enngahlah

Approximately equal to <10,100(4)9>.

E100#^####*#10

quarti-dekahlah

Approximately equal to <10,100(4)10>.

E100#^####*#^####100

deutero-quarticahlah

Approximately equal to <10,100(4)(4)2>. An entire 100^4 tesseract stacked on top of a 100^4 tesseract.

E100#^####*#^####*#^####100

trito-quarticahlah

Approximately equal to <10,100(4)(4)(4)2>.

E100#^####*#^####*#^####*#^####100

teterto-quarticahlah

Approximately equal to <10,100(4)(4)(4)(4)2>.

E100#^#####5

pepto-quarticahlah

Approximately equal to <10,100(4)(4)(4)(4)(4)2>.

E100#^#####6

exto-quarticahlah

Approximately equal to <10,100(4)(4)(4)(4)(4)(4)2>.

E100#^#####7

epto-quarticahlah

Approximately equal to <10,100(4)(4)(4)(4)(4)(4)(4)2>.

E100#^#####8

ogdo-quarticahlah

Approximately equal to <10,100(4)(4)(4)(4)(4)(4)(4)(4)2>.

E100#^#####9

ento-quarticahlah

Approximately equal to <10,100(4)(4)(4)(4)(4)(4)(4)(4)(4)2>.

E100#^#####10

dekato-quarticahlah

Approximately equal to <10,100(4)(4)(4)(4)(4)(4)(4)(4)(4)(4)2>.

E100#^#####20

isosto-quarticahlah

Approximately equal to <10,100(4)(4)(4)(4)(4)(4)(4)(4)(4)(4)(4)(4)(4)(4)(4)(4)(4)(4)(4)(4)2>.

E100#^#####100

quinticahlah / hecato-quarticahlah

Approximately equal to <10,100(5)2>.

E100#^#####100#^#####100

quintitrigahlah

Approximately equal to <10,100(5)3>.

E100#^#####100#^#####100#^#####100

quintitergahlah

Approximately equal to <10,100(5)4>.

E100#^#####*#5

quintipeggahlah

Approximately equal to <10,100(5)5>.

E100#^#####*#6

quintihexgahlah

Approximately equal to <10,100(5)6>.

E100#^#####*#7

quintihepgahlah

Approximately equal to <10,100(5)7>.

E100#^#####*#8

quinti-ahtgahlah

Approximately equal to <10,100(5)8>.

E100#^#####*#9

quinti-enngahlah

Approximately equal to <10,100(5)9>.

E100#^#####*#10

quinti-dekahlah

Approximately equal to <10,100(5)10>.

E100#^#####*#^#####100

deutero-quinticahlah

Approximately equal to <10,100(5)(5)2>.

E100#^#####*#^#####*#^#####100

trito-quinticahlah

Approximately equal to <10,100(5)(5)(5)2>.

E100#^#####*#^#####*#^#####*#^#####100

teterto-quinticahlah

Approximately equal to <10,100(5)(5)(5)(5)2>.

E100#^######5

pepto-quinticahlah

Approximately equal to <10,100(5)(5)(5)(5)(5)2>.

E100#^######6

exto-quinticahlah

Approximately equal to <10,100(5)(5)(5)(5)(5)(5)2>.

E100#^######7

epto-quinticahlah

Approximately equal to <10,100(5)(5)(5)(5)(5)(5)(5)2>.

E100#^######8

ogdo-quinticahlah

Approximately equal to <10,100(5)(5)(5)(5)(5)(5)(5)(5)2>.

E100#^######9

ento-quinticahlah

Approximately equal to <10,100(5)(5)(5)(5)(5)(5)(5)(5)(5)2>.

E100#^######10

dekato-quinticahlah

Approximately equal to <10,100(5)(5)(5)(5)(5)(5)(5)(5)(5)(5)2>.

E100#^######100

sexticahlah

Approximately equal to <10,100(6)2>.

E100#^######100#^######100

sextitrigahlah

Approximately equal to <10,100(6)3>.

E100#^######100#^######100#^######100

sextitergahlah

Approximately equal to <10,100(6)4>.

E100#^######*#5

sextipeggahlah

Approximately equal to <10,100(6)5>.

E100#^######*#6

sextihexgahlah

Approximately equal to <10,100(6)6>.

E100#^######*#7

sextihepgahlah

Approximately equal to <10,100(6)7>.

E100#^######*#8

sexti-ahtgahlah

Approximately equal to <10,100(6)8>.

E100#^######*#9

sexti-enngahlah

Approximately equal to <10,100(6)9>.

E100#^######*#10

sexti-dekahlah

Approximately equal to <10,100(6)10>.

E100#^######*#^######100

deutero-sexticahlah

Approximately equal to <10,100(6)(6)2>.

E100#^######*#^######*#^######100

trito-sexticahlah

Approximately equal to <10,100(6)(6)(6)2>.

E100#^######*#^######*#^######*#^######100

teterto-sexticahlah

Approximately equal to <10,100(6)(6)(6)(6)2>.

E100#^#######5

pepto-sexticahlah

Approximately equal to <10,100(6)(6)(6)(6)(6)2>.

E100#^#######6

exto-sexticahlah

Approximately equal to <10,100(6)(6)(6)(6)(6)(6)2>.

E100#^#######7

epto-sexticahlah

Approximately equal to <10,100(6)(6)(6)(6)(6)(6)(6)2>.

E100#^#######8

ogdo-sexticahlah

Approximately equal to <10,100(6)(6)(6)(6)(6)(6)(6)(6)2>.

E100#^#######9

ento-sexticahlah

Approximately equal to <10,100(6)(6)(6)(6)(6)(6)(6)(6)(6)2>.

E100#^#######10

dekato-sexticahlah

Approximately equal to <10,100(6)(6)(6)(6)(6)(6)(6)(6)(6)(6)2>.

E100#^#######100

septicahlah

Approximately equal to <10,100(7)2>. This number may also be called a hecato-sexticahlah.

E100#^#######100#^#######100

septitrigahlah

Approximately equal to <10,100(7)3>.

E100#^#######100#^#######100#^#######100

septitergahlah

Approximately equal to <10,100(7)4>.

E100#^#######*#5

septipeggahlah

Approximately equal to <10,100(7)5>.

E100#^#######*#6

septihexgahlah

Approximately equal to <10,100(7)6>.

E100#^#######*#7

septihepgahlah

Approximately equal to <10,100(7)7>.

E100#^#######*#8

septi-ahtgahlah

Approximately equal to <10,100(7)8>.

E100#^#######*#9

septi-enngahlah

Approximately equal to <10,100(7)9>.

E100#^#######*#10

septi-dekahlah

Approximately equal to <10,100(7)10>.

E100#^#######*#^#######100

deutero-septicahlah

Approximately equal to <10,100(7)(7)2>.

E100#^#######*#^#######*#^#######100

trito-septicahlah

Approximately equal to <10,100(7)(7)(7)2>.

E100#^#######*#^#######*#^#######*#^#######100

teterto-septicahlah

Approximately equal to <10,100(7)(7)(7)(7)2>.

E100#^########5

pepto-septicahlah

Approximately equal to <10,100(7)(7)(7)(7)(7)2>.

E100#^########6

exto-septicahlah

Approximately equal to <10,100(7)(7)(7)(7)(7)(7)2>.

E100#^########7

epto-septicahlah

Approximately equal to <10,100(7)(7)(7)(7)(7)(7)(7)2>.

E100#^########8

ogdo-septicahlah

Approximately equal to <10,100(7)(7)(7)(7)(7)(7)(7)(7)2>.

E100#^########9

ento-septicahlah

Approximately equal to <10,100(7)(7)(7)(7)(7)(7)(7)(7)(7)2>.

E100#^########10

dekato-septicahlah

Approximately equal to <10,100(7)(7)(7)(7)(7)(7)(7)(7)(7)(7)2>.

E100#^########100

octicahlah

Approximately equal to <10,100(8)2>. This number can also be called hecato-septicahlah.

E100#^########100#^########100

octitrigahlah

Approximately equal to <10,100(8)3>.

E100#^########100#^########100#^########100

octitergahlah

Approximately equal to <10,100(8)4>.

E100#^########*#5

octipeggahlah

Approximately equal to <10,100(8)5>.

E100#^########*#6

octihexgahlah

Approximately equal to <10,100(8)6>.

E100#^########*#7

octihepgahlah

Approximately equal to <10,100(8)7>.

E100#^########*#8

octi-ahtgahlah

Approximately equal to <10,100(8)8>.

E100#^########*#9

octi-enngahlah

Approximately equal to <10,100(8)9>.

E100#^########*#10

octi-dekgahlah

Approximately equal to <10,100(8)10>.

E100#^########*#^########100

deutero-octicahlah

Approximately equal to <10,100(8)(8)2>.

E100#^########*#^########*#^########100

trito-octicahlah

Approximately equal to <10,100(8)(8)(8)2>.

E100#^########*#^########*#^########*#^########100

teterto-octicahlah

Approximately equal to <10,100(8)(8)(8)(8)2>.

E100#^#########5

pepto-octicahlah

Approximately equal to <10,100(8)(8)(8)(8)(8)2>.

E100#^#########6

exto-octicahlah

Approximately equal to <10,100(8)(8)(8)(8)(8)(8)2>.

E100#^#########7

epto-octicahlah

Approximately equal to <10,100(8)(8)(8)(8)(8)(8)(8)2>.

E100#^#########8

ogdo-octicahlah

Approximately equal to <10,100(8)(8)(8)(8)(8)(8)(8)(8)2>.

E100#^#########9

ento-octicahlah

Approximately equal to <10,100(8)(8)(8)(8)(8)(8)(8)(8)(8)2>.

E100#^#########10

dekato-octicahlah

Approximately equal to <10,100(8)(8)(8)(8)(8)(8)(8)(8)(8)(8)2>.

E100#^#########100

nonicahlah

Approximately equal to <10,100(9)2>. This number can also be called hecato-octicahlah.

{10,10(10)2} = 10^10 & 10s

dimendecal

This forms a 10-dimensional cube of edge length 10 all of 10s. It's a little smaller than a decicahlah, since a decicahlah is more like a 10-dimensional cube of edge length 100 all of 10s. A dimendecal has 10,000,000,000 entries!

E100#^##########100

decicahlah

Approximately equal to <10,100(10)2>.

E100#^#^#11

undecicahlah

Approximately equal to <10,100(11)2>.

E100#^#^#12

duodecicahlah

Approximately equal to <10,100(12)2>.

E100#^#^#13

tredecicahlah

Approximately equal to <10,100(13)2>.

E100#^#^#14

quattuordecicahlah

Approximately equal to <10,100(14)2>.

E100#^#^#15

quindecicahlah

Approximately equal to <10,100(15)2>.

E100#^#^#16

sexdecicahlah

Approximately equal to <10,100(16)2>.

E100#^#^#17

septendecicahlah

Approximately equal to <10,100(17)2>.

E100#^#^#18

octodecicahlah

Approximately equal to <10,100(18)2>.

E100#^#^#19

novemdecicahlah

Approximately equal to <10,100(19)2>.

E100#^#^#20

viginticahlah

Approximately equal to <10,100(20)2>.

E100#^#^#30

triginticahlah

Approximately equal to <10,100(30)2>.

E100#^#^#40

quadraginticahlah

Approximately equal to <10,100(40)2>.

E100#^#^#50

quinquaginticahlah

Approximately equal to <10,100(50)2>.

E100#^#^#60

sexaginticahlah

Approximately equal to <10,100(60)2>.

E100#^#^#70

septuaginticahlah

Approximately equal to <10,100(70)2>.

E100#^#^#80

octoginticahlah

Approximately equal to <10,100(80)2>.

E100#^#^#90

nonaginticahlah

Approximately equal to <10,100(90)2>.

{10,10(100)2}

gongulus

One of Bowers' most famous googolisms ... the gongulus. In a way, this combines Jonathan Bowers fascination with large numbers with his fascination of higher dimensions. This is a 100-dimensional cube of entries, of edge length 10 all of 10s. There would literally be a googol (10^100) entries in this expression. This means we would not be able to write it out in full. This number is only slightly smaller than a godgather, which brings us to ...

E100#^#^#100

godgathor

            A godgathor is comparable to Jonathan Bowers' gongulus. It's comparable to a 100-dimensional array. Evaluating godgathor we obtain:

E100#^#^#100 = E100#^#####...#####100 w/100 #s in a row

A godgathor should be greater than <10,100(100)2> which is larger than <10,10(100)2>. Thus a godgathor should be slightly larger than a gongulus. <10,100(100)2> is also equivalent to <10,100(0,1)2>.

E100#^#101100

goober bunch

This number was coined by me to break up the monotony of my "main sequence" numbers. It can be defined as E100(#^#^100)^#100. It was intended to illustrate that #^#^# is far worse than merely #^#^100. The topmost hyperion can be replaced by any positive integer that ExE can generate. So E100#^#^#*#^#^#100 for example is much much worse than simply E100#^#^(100)*#^#^(100)100. Only the second one get's substituted. Thus we have E100#^#^#*#^#^#100 = E100#^#^#*#^#^(100)100. #^#^(100) is then used to create godgathor-level recursion on top of #^#^#. When it's done we have #^#^N where N is waaaaaaaayyyyyyyy larger than a mere 100. A goober bunch looks like it should be huge relative to something as modest as E100#^#^#100#100, for example, but in fact E100#^#^#100#100 is way way larger than a mere goober bunch.

{10,10({10,10(100)2})2}

gongulusplex

Now here is a cool number. It's a gongulus-dimensional cube of edge length 10 all of 10s. It would have a 10^gongulus entries, a gongulus-plexed entries if you will :p

This passes up the godgathor of course, but it's smaller than ...

E100#^#^#100#2

grand godgathor

Yes we still can stack grands on these new numbers, and they always work the same. We just append as #2, or a #(n+1) if it already ends in #n.

E100#^#(1+E100#^#^(101)100)(E100#^#101100)

gibbering goober bunch

Sorta like the "grand" version of goober bunch, but highly unorthodox. Despite how "large" it might seem it is still smaller than ...

E100#^#^#100#3

grand grand godgathor

The "double grand" version of godgathor. Now let's move on to ...

E100#^#^#100#^#^#100

gotrigathor

Yup. We can use the infix multipliers here too. This is conjectured to be approximately equal to <10,100(0,1)3>.

E100#^#^#100#^#^#100#^#^#100

gotergathor

This is conjectured to be approximately equal to <10,100(0,1)4>.

E100#^#^#*#5

gopeggathor

This is conjectured to be approximately equal to <10,100(0,1)5>.

E100#^#^#*#6

gohexgathor

This is conjectured to be approximately equal to <10,100(0,1)6>.

E100#^#^#*#7

gohepgathor

This is conjectured to be approximately equal to <10,100(0,1)7>.

E100#^#^#*#8

go-ahtgathor

This is conjectured to be approximately equal to <10,100(0,1)8>.

E100#^#^#*#9

go-enngathor

This is conjectured to be approximately equal to <10,100(0,1)9>.

E100#^#^#*#10

godekgathor

This is conjectured to be approximately equal to <10,100(0,1)10>.

E100#^#^#*#^#^#100

deutero-godgathor

This is conjectured to be approximately equal to <10,100(0,1)(0,1)2>.

E100#^#^#*#^#^#*#^#^#100

trito-godgathor

This is conjectured to be approximately equal to <10,100(0,1)(0,1)(0,1)2>.

E100#^#^#*#^#^#*#^#^#*#^#^#100

teterto-godgathor

This is conjectured to be approximately equal to <10,100(0,1)(0,1)(0,1)(0,1)2>.

E100#^(#^#*#)5

pepto-godgathor

This is conjectured to be approximately equal to <10,100(0,1)(0,1)(0,1)(0,1)(0,1)2>.

E100#^(#^#*#)6

exto-godgathor

This is conjectured to be approximately equal to <10,100(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)2>.

E100#^(#^#*#)7

epto-godgathor

This is conjectured to be approximately equal to <10,100(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)2>.

E100#^(#^#*#)8

ogdo-godgathor

This is conjectured to be approximately equal to <10,100(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)2>.

E100#^(#^#*#)9

ento-godgathor

This is conjectured to be approximately equal to <10,100(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)2>.

E100#^(#^#*#)10

dekato-godgathor

This is conjectured to be approximately equal to <10,100(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)(0,1)2>.

E100#^(#^#*#)20

isosto-godgathor

This is conjectured to be approximately equal to <10,100(0,1)...(0,1)2>. w/20 (0,1)'s.

E100#^(#^#*#)100

hecato-godgathor

Approximately equal to <10,100(1,1)2>. This is also equivalent to <10,100(0,1)...(0,1)2> w/100 (0,1)'s.

E100#^(#^#*##)100

godgridgathor

Approximately equal to <10,100(2,1)2>. A 2-D array of 100-D arrays. More precisely it's a 100^2 array of 100^100 arrays.

{3,3(0,2)2}

dulatri

One of the original 120 named numbers by Jonathan Bowers. The notation {3,3(0,2)2} is modern and didn't exist back in 2002 when Bowers' first introduced this number. Originally it Bowers' defined it as a "(3^3)^2 array of 3's". Imagine a 3^3 array of 3^3 arrays. This can be seen by expanding the expression to the following:

{ 3^3&3 (0,1) 3^3&3 (0,1) 3^3&3 (1,1) 3^3&3 (0,1) 3^3&3 (0,1) 3^3&3 (1,1) 3^3&3 (0,1) 3^3&3 (0,1) 3^3&3 (2,1) 3^3&3 (0,1) 3^3&3 (0,1) 3^3&3 (1,1) 3^3&3 (0,1) 3^3&3 (0,1) 3^3&3 (1,1) 3^3&3 (0,1) 3^3&3 (0,1) 3^3&3 (2,1) 3^3&3 (0,1) 3^3&3 (0,1) 3^3&3 (1,1) 3^3&3 (0,1) 3^3&3 (0,1) 3^3&3 (1,1) 3^3&3 (0,1) 3^3&3 (0,1) 3^3&3 }

The "separators" here, are not themselves denoting additional "dimensions", at least not in the usual sense. Instead (0,1) separates any dimensional spaces. So the number of dimensions between (0,1) are not limited to 3, but any finite dimensional space. With that in mind, we can have "rows of dimensions" and "planes of dimensions" and so on. This can be thought of as going beyond infinite dimensional space, and continuing to (w+1)-dimensional space, (w+2)-dimensional space, and so on.

E100#^(#^#*###)100

godkubikgathor

Approximately equal to <10,100(3,1)2>. A 3-D array of 100-D arrays. More precisely it's a 100^3 array of 100^100 arrays.

E100#^(#^#*####)100

godquarticgathor

Approximately equal to <10,100(4,1)2>. A 4-D array of 100-D arrays.

E100#^(#^#*#####)100

godquinticgathor

Approximately equal to <10,100(5,1)2>. A 5-D array of 100-D arrays.

E100#^(#^#*######)100

godsexticgathor

Approximately equal to <10,100(6,1)2>. A 6-D array of 100-D arrays.

E100#^(#^#*#######)100

godsepticgathor

Approximately equal to <10,100(7,1)2>. A 7-D array of 100-D arrays.

E100#^(#^#*########)100

godocticgathor

Approximately equal to <10,100(8,1)2>. A 8-D array of 100-D arrays.

E100#^(#^#*#########)100

godnonicgathor

Approximately equal to <10,100(9,1)2>. A 9-D array of 100-D arrays.

E100#^(#^#*##########)100

goddecicgathor

Approximately equal to <10,100(10,1)2>. A 10-D array of 100-D arrays.

E100#^(#^#*#^#)100

godgathordeus

Approximately equal to <10,100(0,2)2> = <10,100(100,1)2>. This would be much larger than a dulatri even though this is also a dimensional array of dimensional arrays. However in this case it's a 100-dimensional array of 100-dimensional arrays.

E100#^(#^#*#^#*#)100

hecato-godgathordeus

Approximately equal to <10,100(1,2)2>.

E100#^(#^#*#^#*##)100

godgridgathordeus

Approximately equal to <10,100(2,2)2>.

E100#^(#^#*#^#*###)100

godkubikgathordeus

Approximately equal to <10,100(3,2)2>.

E100#^(#^#*#^#*####)100

godquarticgathordeus

Approximately equal to <10,100(4,2)2>.

E100#^(#^#*#^#*#####)100

godquinticgathordeus

Approximately equal to <10,100(5,2)2>.

E100#^(#^#*#^#*######)100

godsexticgathordeus

Approximately equal to <10,100(6,2)2>.

E100#^(#^#*#^#*#######)100

godsepticgathordeus

Approximately equal to <10,100(7,2)2>.

E100#^(#^#*#^#*########)100

godocticgathordeus

Approximately equal to <10,100(8,2)2>.

E100#^(#^#*#^#*#########)100

godnonicgathordeus

Approximately equal to <10,100(9,2)2>.

E100#^(#^#*#^#*##########)100

goddecicgathordeus

Approximately equal to <10,100(10,2)2>.

E100#^(#^#*#^#*#^#)100

godgathortruce

Approximately equal to <10,100(0,3)2> = <10,100(100,2)2>. Let's introduce the rest of the postfix operators, beginning with -truce.

E100#^(#^#*#^#*#^#*#^#)100

godgathorquad

Approximately equal to <10,100(0,4)2>.

E100#^(#^#*#^#*#^#*#^#*#^#)100

godgathorquid

Approximately equal to <10,100(0,5)2>.

E100#^(#^#*#^#*#^#*#^#*#^#*#^#)100

godgathorsid

Approximately equal to <10,100(0,6)2>.

E100#^#^##7

godgathorseptuce

Approximately equal to <10,100(0,7)2>.

E100#^#^##8

godgathoroctuce

Approximately equal to <10,100(0,8)2>.

E100#^#^##100

gralgathor

Approximately equal to <10,100(0,0,1)2>. Here we introduce the new "gral" root for the first time. This indicates that the uppermost exponent of the hyperion power tower is '##'.

E100#^#^##100#^#^##100

graltrigathor

Approximately equal to <10,100(0,0,1)3>.

E100#^#^##100#^#^##100#^#^##100

graltergathor

Approximately equal to <10,100(0,0,1)4>.

E100#^#^##*#^#^##100

deutero-gralgathor

Approximately equal to <10,100(0,0,1)(0,0,1)2>.

E100#^#^##*#^#^##*#^#^##100

trito-gralgathor

Approximately equal to <10,100(0,0,1)(0,0,1)(0,0,1)2>.

E100#^#^##*#^#^##*#^#^##*#^#^##100

teterto-gralgathor

Approximately equal to <10,100(0,0,1)(0,0,1)(0,0,1)(0,0,1)2>.

E100#^(#^##*#^##)100

gralgathordeus

Approximately equal to <10,100(0,0,2)2>.

{3,3(0,0,0,1)2}

trimentri

One of Bowers' original 120 named numbers. This array is also equivalent to {3,3(0,0,3)2}, which is clearly smaller than a gralgathortruce but larger than gralgathordeus.

E100#^(#^##*#^##*#^##)100

gralgathortruce

Approximately equal to <10,100(0,0,3)2>.

E100#^(#^##*#^##*#^##*#^##)100

gralgathorquad

Approximately equal to <10,100(0,0,4)2>.

E100#^#^###5

gralgathorquid

Approximately equal to <10,100(0,0,5)2>.

E100#^#^###6

gralgathorsid

Approximately equal to <10,100(0,0,6)2>.

E100#^#^###7

gralgathorseptuce

Approximately equal to <10,100(0,0,7)2>.

E100#^#^###8

gralgathoroctuce

Approximately equal to <10,100(0,0,8)2>.

E100#^#^###100

thraelgathor

Approximately equal to <10,100(0,0,0,1)2>.

E100#^#^###100#^#^###100

thraeltrigathor

Approximately equal to <10,100(0,0,0,1)3>.

E100#^#^###*#^#^###100

deutero-thraelgathor

Approximately equal to <10,100(0,0,0,1)(0,0,0,1)2>.

E100#^#^###*#^#^###*#^#^###100

trito-thraelgathor

Approximately equal to <10,100(0,0,0,1)(0,0,0,1)(0,0,0,1)2>.

E100#^(#^###*#^###)100

thraelgathordeus

Approximately equal to <10,100(0,0,0,2)2>.

E100#^(#^###*#^###*#^###)100

thraelgathortruce

Approximately equal to <10,100(0,0,0,3)2>.

E100#^#^####100

terinngathor

Approximately equal to <10,100(0,0,0,100)2> = <10,100(0,0,0,0,1)2>.

E100#^#^####100#^#^####100

terinntrigathor

Approximately equal to <10,100(0,0,0,0,1)3>.

E100#^#^####*#^#^####100

deutero-terinngathor

Approximately equal to <10,100(0,0,0,0,1)(0,0,0,0,1)2>.

E100#^#^####*#^#^####*#^#^####100

trito-terinngathor

Approximately equal to <10,100(0,0,0,0,1)(0,0,0,0,1)(0,0,0,0,1)2>.

E100#^(#^####*#^####)100

terinngathordeus

Approximately equal to <10,100(0,0,0,0,2)2>.

E100#^(#^####*#^####*#^####)100

terinngathortruce

Approximately equal to <10,100(0,0,0,0,3)2>.

E100#^#^#####100

pehaelgathor

Approximately equal to <10,100(0,0,0,0,100)2> = <10,100(0,0,0,0,0,1)2>. 

E100#^#^######100

hexaelgathor

Approximately equal to <10,100(0,0,0,0,0,100)2> = <10,100(0,0,0,0,0,0,1)2>.

E100#^#^#######100

heptaelgathor

Approximately equal to <10,100(0,0,0,0,0,0,100)2> = <10,100(0,0,0,0,0,0,0,1)2>.

E100#^#^########100

octaelgathor

Approximately equal to <10,100(0,0,0,0,0,0,0,100)2> = <10,100(0,0,0,0,0,0,0,0,1)2>.

E100#^#^#^#100

godtothol

Approximately equal to <10,100(0,0,...,0,0,1)2> w/100 0s. This can also be simplified to <10,100((1)1)2>. At this point we have reached X^X^X arrays. However we are still only getting started. Now we can start climbing our way up to higher and higher power towers of #'s.

E100#^#^#^##100

graltothol

Conjectured to be approximately <10,100((2)1)2>. These googolism's are composed of two parts. The first tells us how many hyperions are at the top of the power tower, and the second tells us how many levels up it is until the top.

E100#^#^#^###100

thraeltothol

Conjectured to be approximately <10,100((3)1)2>.

E100#^#^#^####100

terinntothol

Conjectured to be approximately <10,100((4)1)2>.

E100#^#^#^#####100

pehaeltothol

Conjectured to be approximately <10,100((5)1)2>.

E100#^#^#^######100

hexaeltothol

Conjectured to be approximately <10,100((6)1)2>.

E100#^#^#^#######100

heptaeltothol

Conjectured to be approximately <10,100((7)1)2>.

E100#^#^#^########100

octaeltothol

Conjectured to be approximately <10,100((8)1)2>.

E100#^#^#^#^#100

godtertol

Approximately equivalent to X^X^X^X arrays. Conjectured to be approximately equal to <10,100((100)1)2> = <10,100((0,1)1)2>. The root here "tertol" implies "4", and this is because there are 4 #'s below the topmost exponent.

E100#^#^#^#^##100

graltertol

Conjectured to be approximately equal to <10,100((0,0,1)1)2>.

E100#^#^#^#^###100

thraeltertol

Conjectured to be approximately equal to <10,100((0,0,0,1)1)2>.

E100#^#^#^#^####100

terinntertol

Conjectured to be approximately equal to <10,100((0,0,0,0,1)1)2>.

E100#^#^#^#^#####100

pehaeltertol

Conjectured to be approximately equal to <10,100((0,0,0,0,0,1)1)2>.

E100#^#^#^#^######100

hexaeltertol

Conjectured to be approximately equal to <10,100((0,0,0,0,0,0,1)1)2>.

E100#^#^#^#^#######100

heptaeltertol

Conjectured to be approximately equal to <10,100((0,0,0,0,0,0,0,1)1)2>.

E100#^#^#^#^########100

octaeltertol

Conjectured to be approximately equal to <10,100((0,0,0,0,0,0,0,0,1)1)2>.

E100#^#^#^#^#^#100

godtopol

Approximately equivalent to X^^5 arrays. Conjectured to be approximately equal to <10,100(((1)1)1)2>. The root "topol" here is chosen for a bit of fun. Instead of the perhaps more predictable "peptol" or something of that sort, I opted to use a root that suggests "topple" as this power tower is at risk of toppling over at any moment now!

E100#^#^#^#^#^##100

graltopol

Conjectured to be approximately equal to <10,100(((2)1)1)2>.

E100#^#^#^#^#^###100

thraeltopol

Conjectured to be approximately equal to <10,100(((3)1)1)2>.

E100#^#^#^#^#^####100

terinntopol

Conjectured to be approximately equal to <10,100(((4)1)1)2>.

E100#^#^#^#^#^#####100

pehaeltopol

Conjectured to be approximately equal to <10,100(((5)1)1)2>.

E100#^#^#^#^#^######100

hexaeltopol

Conjectured to be approximately equal to <10,100(((6)1)1)2>.

E100#^#^#^#^#^#######100

heptaeltopol

Conjectured to be approximately equal to <10,100(((7)1)1)2>.

E100#^#^#^#^#^########100

octaeltopol

Conjectured to be approximately equal to <10,100(((8)1)1)2>.

E100#^#^#^#^#^#^#100

godhathor

Approximately equivalent to X^^6 arrays. Conjectured to be approximately equal to <10,100(((0,1)1)1)2>. Again, rather than use something like "extol" (it was actually originally extathol), I used "hathor", as it ryhmns with "gathor", is the name of an egyptian god (a theme will see a little more of later on), and also starts with "h" like "hex". 

E100#^#^#^#^#^#^##100

gralhathor

Conjectured to be approximately equal to <10,100(((0,0,1)1)1)2>.

E100#^#^#^#^#^#^###100

thraelhathor

Conjectured to be approximately equal to <10,100(((0,0,0,1)1)1)2>.

E100#^#^#^#^#^#^####100

terinnhathor

Conjectured to be approximately equal to <10,100(((0,0,0,0,1)1)1)2>.

E100#^#^#^#^#^#^#####100

pehaelhathor

Conjectured to be approximately equal to <10,100(((0,0,0,0,0,1)1)1)2>.

E100#^#^#^#^#^#^######100

hexaelhathor

Conjectured to be approximately equal to <10,100(((0,0,0,0,0,0,1)1)1)2>.

E100#^#^#^#^#^#^#######100

heptaelhathor

Conjectured to be approximately equal to <10,100(((0,0,0,0,0,0,0,1)1)1)2>.

E100#^#^#^#^#^#^########100

octaelhathor

Conjectured to be approximately equal to <10,100(((0,0,0,0,0,0,0,0,1)1)1)2>.

E100#^#^#^#^#^#^#^#100

godheptol

Approximately equivalent to X^^7 arrays. Conjectured to be approximately equal to <10,100((((1)1)1)1)2>.

E100#^#^#^#^#^#^#^##100

gralheptol

Conjectured to be approximately equal to <10,100((((2)1)1)1)2>.

E100#^#^#^#^#^#^#^###100

thraelheptol

Conjectured to be approximately equal to <10,100((((3)1)1)1)2>.

E100#^#^#^#^#^#^#^####100

terinnheptol

Conjectured to be approximately equal to <10,100((((4)1)1)1)2>.

E100#^#^#^#^#^#^#^#####100

pehaelheptol

Conjectured to be approximately equal to <10,100((((5)1)1)1)2>.

E100#^#^#^#^#^#^#^######100

hexaelheptol

Conjectured to be approximately equal to <10,100((((6)1)1)1)2>.

E100#^#^#^#^#^#^#^#######100

heptaelheptol

Conjectured to be approximately equal to <10,100((((7)1)1)1)2>.

E100#^#^#^#^#^#^#^########100

octaelheptol

Conjectured to be approximately equal to <10,100((((8)1)1)1)2>.

E100#^#^#^#^#^#^#^#^#100

godoctol

Approximately equivalent to X^^8 arrays. Conjectured to be approximately equal to <10,100((((0,1)1)1)1)2>.

E100#^#^#^#^#^#^#^#^##100

graloctol

Conjectured to be approximately equal to <10,100((((0,0,1)1)1)1)2>.

E100#^#^#^#^#^#^#^#^###100

thraeloctol

Conjectured to be approximately equal to <10,100((((0,0,0,1)1)1)1)2>.

E100#^#^#^#^#^#^#^#^####100

terinnoctol

Conjectured to be approximately equal to <10,100((((0,0,0,0,1)1)1)1)2>.

E100#^#^#^#^#^#^#^#^#####100

pehaeloctol

Conjectured to be approximately equal to <10,100((((0,0,0,0,0,1)1)1)1)2>.

E100#^#^#^#^#^#^#^#^######100

hexaeloctol

Conjectured to be approximately equal to <10,100((((0,0,0,0,0,0,1)1)1)1)2>.

E100#^#^#^#^#^#^#^#^#######100

heptaeloctol

Conjectured to be approximately equal to <10,100((((0,0,0,0,0,0,0,1)1)1)1)2>.

E100#^#^#^#^#^#^#^#^########100

octaeloctol

Conjectured to be approximately equal to <10,100((((0,0,0,0,0,0,0,0,1)1)1)1)2>.

E100#^^#100
tethrathoth

            The tethrathoth (or simply thoth) is equal to E100#^#^#^#^ ... ^#^#^#^#100 w/100 #s. It is comparable and slightly smaller than Jonathan Bowers' goppatoth. It is roughly equivalent to an X^^99 array. It is conjectured that this number is approximately:

<10,100((((((((((((((((((((((((((((((((((((((((((((((((((1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)2>

Using Bowers' Array Notation. At this point we've reached the practical limits of Bowers separator notation, as he does not officially have a way to directly write out arrays beyond this point. This leads us to ...

10^^100 & 10
goppatoth

One of Bowers' largest of his original set of 121 googolisms. This one is interesting in being at the limit of what is considered to be well defined in array notation (even to this day!). This would be a tetrational array. Specifically it would be a X^^100 array, making it just slightly larger than a tethrathoth. In fact, a goppatoth should be approximately E100#^^#101 in ExE. Goppatoth is the first number that Bowers does not write out directly in array notation, even though it is in fact possible. Instead he only describes the "array structure" using his array structure shorthand. Even though Bowers' does not provide an exact array we can write it out ourselves as:

<10,10((((((((((((((((((((((((((((((((((((((((((((((((((0,1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)1)2>

This, when expanded out completely, will give us 10^^100 entries all of 10. At this point the separator notation is bursting at the seams and is not particularly useful. It should also be noted that this is not the same as a size 100 X^^X array of 10s. That would have 100^^100 entries all of 10s. This quirk of array notation is what makes a lot of these late Bowerisms difficult to express directly.

E100,000#^^#100,000

tethrathothigong

            For old time sake, here is the "gong" version of a tethrathoth, the tethrathothigong. It is equal to E100,000#^#^#^#^...^#^#^#^#100,000 w/100,000 #s. In array notation is would be a whopping X^^99,999 array! Technically speaking this number is mind-crushingly larger than having a mere hundred cascade levels ... but it's still a very modest improvement compare to how far we're going to take it from here ...

E100#^^#100#2

grand tethrathoth

            The grand tethrathoth is equal to E100#^#^#^#^ ... ^#^#^#^#100 w/tethrathoth #s. This is incomparably larger than the tethrathothigong, though from a googologist's perspective just a stone's skip away from the tethrathoth. It is comparable to and slightly smaller than Jonathan Bowers' goppatothplex. It is roughly equivalent to a X^^(tethrathoth-1) array. The next explicitly named googolism after a goppatothplex is a triakulus. This number is much much MUCH larger and it will take us a long time to reach it!

10^^goppatoth & 10

goppatothplex

A X^^goppatoth array. This makes it just "slightly" larger than a grand tethrathoth. This is notable for being the last well-defined googolism of Bowers' original set. The next Bowerism of the original set is a tridecatrix, which is {10,10,10} & 10. However pentational arrays have never been formally defined, let along dekational arrays! These will be added for size reference regardless, based on there theoretical placement. 

E100,000#^^#100,000#2

grand tethrathothigong

The "grand" version of a tethrathothigong would be much larger than a goppatothplex but way way smaller than a ...

E100#^^#100#3

grand grand tethrathoth

Even without new names we can repeatedly apply grand to create an arbitrary number of new "googolisms". Thus there is never really any artificial stopping point.

E100#^^#100#4

grand grand grand tethrathoth

Let's keeping adding grands and see where that leads ...

E100#^^#100#5

grand grand grand grand tethrathoth

4 grands ...

E100#^^#100#6

grand grand grand grand grand tethrathoth

5 grands ...

E100#^^#100#7

grand grand grand grand grand grand tethrathoth

6 grands ... skip a few ...

E100#^^#100#100

ninety-nine-ex-grand tethrathoth / grantethrathoth

99 grands ... (we can call this one grantethrathoth from grangol + tethrathoth) ...

E100#^^#100#101

hundred-ex-grand tethrathoth

100 grands ... now what ... hmm ...

E100#^^#100#(1+E100)

googol-ex-grand tethrathoth

... a googol grands ...

E100#^^#100#(1+E100#2)

googolplex-ex-grand tethrathoth

... a googolplex grands ...

E100#^^#100#(1+E100#100)

grangol-ex-grand tethrathoth

... a grangol grands ...

E100#^^#100#(1+E100##100)

gugold-ex-grand tethrathoth

... a gugold grands ...

E100#^^#100#(1+E100#^#100)

godgahlah-ex-grand tethrathoth

... a godgahlah grands ... 'member godgahlah ... seems quaint now doesn't it ...

E100#^^#100#1#2

tethrathoth-minus-one-ex-grand tethrathoth

Here we can use are simple "-ex-grand" construction to name some ExE expressions.

E100#^^#100#(1+E100#^^#100)

tethrathoth-ex-grand tethrathoth

Note that having a tethrathoth grand's is not the same as E100#^^#100#1#2. There is an offset of 1 that has to be accounted for, annoyingly.

E100#^^#100#2#2

grand-tethrathoth-minus-one-ex-grand tethrathoth

Another ExE expression which can be awkwardly named using the "-ex-grand" construct.

E100#^^#100#(1+E100#^^#100#2)

grand-tethrathoth-ex-grand tethrathoth

Again grand-tethrathoth-ex-grand tethrathoth != E100#^^#100#2#2, even though it is close.

E100#^^#100#3#2

grand-grand-tethrathoth-minus-one-ex-grand tethrathoth

We can keep making these constructions fairly easily.

E100#^^#100#(1+E100#^^#100#3)

grand-grand-tethrathoth-ex-grand tethrathoth

grand-grand-tethrathoth-ex-grand tethrathoth != E100#^^#100#3#2.

E100#^^#100#4#2

grand-grand-grand-tethrathoth-minus-one-ex-grand tethrathoth

E100#^^#100#4#2 = E100#^^#100#(E100#^^#100#4) which is smaller than ...

E100#^^#100#(1+E100#^^#100#4)

grand-grand-grand-tethrathoth-ex-grand tethrathoth

grand-grand-grand-tethrathoth-ex-grand tethrathoth > E100#^^#100#4#2

E100#^^#100#5#2

grand-grand-grand-grand-tethrathoth-minus-one-ex-grand tethrathoth

E100#^^#100#5#2 = E100#^^#100#(E100#^^#100#5) which is smaller than ...

E100#^^#100#(1+E100#^^#100#5)

grand-grand-grand-grand-tethrathoth-ex-grand tethrathoth

grand-grand-grand-grand-tethrathoth-ex-grand tethrathoth > E100#^^#100#5#2.

E100#^^#100#100#2

grand grantethrathoth

Much larger than any of the previous entries is the grand version of a grantethrathoth. This is vanishingly small compare to ...

E100#^^#100#1#3

tethrathoth-minus-one-ex-grand-tethrathoth-minus-one-ex-grand tethrathoth

E100#^^#100#1#3 = E100#^^#100#(E100#^^#100#(E100#^^#100)) = E100#^^#100#(E100#^^#100)#2. The names starting from here are getting rather cumbersome...

E100#^^#100#(1+E100#^^#100#(1+E100#^^#100))

tethrathoth-ex-grand-tethrathoth-ex-grand tethrathoth

As you can see, nesting the "ex-grand" construct does allow us to get even further. But how much progress are we really making against ExE notation?

E100#^^#100#2#3

Not much. This expands to E100#^^#100#(E100#^^#100#(E100#^^#100#2)) which naturally is much much larger. Those +1s can not compete.

E100#^^#100#(1+E100#^^#100#(1+E100#^^#100#2))

grand-tethrathoth-ex-grand-tethrathoth-ex-grand tethrathoth

This will in turn be beaten by ...

E100#^^#100#3#3

Which is still relatively small.

E100#^^#100#(1+E100#^^#100#(1+E100#^^#100#3))

grand-grand-tethrathoth-ex-grand-tethrathoth-ex-grand tethrathoth

Two -ex-grands ... we must be getting far now ...

E100#^^#100#100#3

grand grand grantethrathoth

Actually we are practically standing still. We are still only up to the grand grand grantethrathoth.

E100#^^#100#1#4

This number is larger than E100#^^#100#100#3, despite the 1. Simply expand it using the variant recursive rule: E100#^^#100#1#4 = E100#^^#100#(E100#^^#100)#3 > E100#^^#100#100#3.

E100#^^#100#(1+E100#^^#100#(1+E100#^^#100#(1+E100#^^#100)))

tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand tethrathoth

Three applications of "-ex-grand" yet it's still smaller than ...

E100#^^#100#100#4

grand grand grand grantethrathoth

It should be clear now that nesting "-ex-grand" is only going to get us about two elementary recursions above a tethrathoth. Like before this number is still smaller than ...

E100#^^#100#1#5

E100#^^#100#1#5 = E100#^^#100#(E100#^^#100)#4 > E100#^^#100#100#4.

E100#^^#100#(1+E100#^^#100#(1+E100#^^#100#(1+E100#^^#100#(1+E100#^^#100))))

tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand tethrathoth

Larger than E100#^^#100#1#5 and yet much much smaller than...

E100#^^#100#100#5

grand grand grand grand grantethrathoth

From here it should be obvious that E100#^^#100#100#n is always less than E100#^^#100#1#(n+1). Therefore this number is less than E100#^^#100#1#6 which is much much less than ...

E100#^^#100#1#100

This number is less than ...

E100#^^#100#(1+E100#^^#100#( ... ... (1+E100#^^#100)) ... ... ))

w/100 E100#^^#100s

tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand-tethrathoth-ex-grand tethrathoth

A modest attempt to extend the tethrathoth using the obvious approach of extending the sequence: tethrathoth, grand tethrathoth, grand grand tethrathoth, three-ex-grand tethrathoth, four-ex-grand tethrathoth, ... etc. This extension however is not as ambitious as it seems! This number is still smaller than a greatethrathoth. We've really only begun. Even without inventing a single new delimiter, we have only begun. We can do with much more even with just #^^# and the exisiting ExE.

E100#^^#100#100#100

greatethrathoth

greagol + tethrathoth = greathrathoth. This is still less than E100#^^#100#1#101 which is much much smaller than ...

E100#^^#100#1#1#2

E100#^^#100#1#1#2 = E100#^^#100#1#(E100#^^#100). This number is smaller than ... 

E100#^^#100#(1+E100#^^#100#( ... ... (1+E100#^^#100)) ... ... ))

w/tethrathoth E100#^^#100s

tethrathoth-ex-grand-tethrathoth-ex-grand- ... ... -tethrathoth-ex-grand tethrathoth

w/tethrathoth "tethrathoth"s

            How crazy is that?! This is the sort of number someone might come up with the extend the tethrathoth series given the notation N-ex-grand tethrathoth is the (N+1)th member of the tethrathoth series. Yet it barely taps into the possibilities with Extended Cascading-E Notation (xE^) or even mere LECEN (Limited-Extension Cascading-E Notation). It's still smaller than a grand greatethrathoth.

E100#^^#100#100#100#2

grand greatethrathoth

Another standard ExE googolism. A modest extension to tethrathoth as far as ExE is concerned. Note:

E100#^^#100#100#100#2 = E100#^^#100#100#(E100#^^#100#100#100).

So grand greatethrathoth is the greatethrathoth member of the grantethrathoth sequence, just like greagolthrex is the greagolth member of the grangol sequence. Yup. That does sound insane. 

E100#^^#100#1#1#3

E100#^^#100#1#1#3 = E100#^^#100#1#(E100#^^#100#1#1#2) > E100#^^#100#1#(1+E100#^^#100#100#100) > E100#^^#100#100#(E100#^^#100#100#100). This number is still smaller than ...

E100#^^#100#(1+E100#^^#100#( ... ... (1+E100#^^#100)) ... ... ))

w/E100#^^#100#(1+E100#^^#100#(...(1+E100#^^#100))...)) "E100#^^#100"s 

w/tethrathoth "E100#^^#100"s

tethrathoth-ex-grand-tethrathoth-ex-grand- ... ... -tethrathoth-ex-grand tethrathoth

w/tethrathoth-ex-grand-...-tethrathoth-ex-grand tethrathoth "tethrathoth"s

w/tethrathoth "tethrathoth"s

            This is just insane!!! Yet this is clearly the wrong approach because even torturing language this far doesn't get us as far as a mere grand grand greatethrathoth. This again shows the linguistic approach is vastly inferior to the mathematical approach.

E100#^^#100#100#100#3

grand grand greatethrathoth

stacking grands on a greathethrathoth. Moving along ...

E100#^^#100#100#100#100

gigantethrathoth

gigangol + tethrathoth. The constructions here should be straight forward. We could run through them, but I've got one last extension for the "ex-grand" construction. In a way we are already a long way from a tethrathoth. The sheer height of the hash-tower is unimaginable now. It's already in roughly the range of the numbers we are currently considering, rendering the height and the number it describes as nearly indistinguishable ... and yet ... we're still only getting started, even with LECEN (Limited Extension Cascading-E Notation).

E100#^^#100#1#1#1#2

Seemingly unimportant, but this number is actually a lower bound on the next entry. The name of this number would be extremely cumbersome. It would basically be the next entry but throw in minus-one's everywhere. As for showing this is larger than the previous entry it is sufficient to expand it to E100#^^#100#1#1#1#2 = E100#^^#100#1#1#tethrathoth > E100#^^#100#100#100#100.

tethrathoth-ex- ... ... -ex-grand-tethrathoth-ex-grand tethrathoth

w/tethrathoth-ex- ... ... -ex-grand-tethrathoth-ex-grand tethrathoth "tethrathoth"s

w/tethrathoth-ex- ... ... -ex-grand-tethrathoth-ex-grand tethrathoth "tethrathoth"s

...

...

...

w/tethrathoth-ex- ... ... -ex-grand-tethrathoth-ex-grand tethrathoth "tethrathoth"s

w/tethrathoth-ex- ... ... -ex-grand-tethrathoth-ex-grand tethrathoth "tethrathoth"s

w/tethrathoth "tethrathoth"s

where there are a tethrathoth lines

            Despite how insane this is, it's still doesn't get one very far. This is still less than a grand gigantethrathoth.

E100#^^#100#100#100#100#2

grand gigantethrathoth

We've gotten surprisingly nowhere with all those "-ex-grands". Let's now unleash the full power of LECEN ...

E100#^^#100#100#100#100#100

gorgetethrathoth

gorgegol + tethrathoth.

E100#^^#100#100#100#100#100#100

gultethrathoth

gulgol + tethrathoth.

E100#^^#100#100#100#100#100#100#100

gasptethrathoth

gaspgol + tethrathoth.

E100#^^#100#100#100#100#100#100#100#100

ginortethrathoth

ginorgol + tethrathoth.

E100#^^#100##9

gargantethrathoth

gargantuul + tethrathoth.

E100#^^#100##10

googontethrathoth

googondol + tethrathoth.

E100#^^#100##100

gugolda-carta-tethrathoth / gugoldatethrathoth

We could simply call this number a gugoldatethrathoth, but eventually this adapting process of smashing googlisms together is going to get messy and inconsistent. Here let's introduce one of the more modern features of ExE that was introduced during the xE^ phase of development. The ability to "map" previous googolism's into new ones would be useful so that we don't have to invent a new rule everytime we want to smash two googolism's together. For this purpose the "carta" infix was created. G-carta-H literally means G mapped to H. More concretely it means stacking G level recursion on top of H. This allows for a wealth of new possibilities.

E100#^^#100###100

throogola-carta-tethrathoth / throogolatethrathoth

Skipping on ahead a bit, we have the throogolatethrathoth or throogola-carta-tethrathoth. Note the use of the extension "a" at the end of throogol since we are connecting it with other elements.

E100#^^#100####100

teroogola-carta-tethrathoth / teroogolatethrathoth

Will be using the "carta" infix exclusively from here on out. However it's not hard to imagine the continuation for the rest of the xE# numbers.

E100#^^#100#^#100

godgahlah-carta-tethrathoth

Here we combine the milestones of the end of xE# and E^ respectively. It will be a long while before we reach the limit of xE^.

E100#^^#100#^#100#^#100

gotrigahlah-carta-tethrathoth

We can combine our previous tricks with the carta infix.

E100#^^#100#^#*#100

godgoldgahlah-carta-tethrathoth

godgoldgahlah was a long while ago now.

E100#^^#100#^#*##100

godthroogahlah-carta-tethrathoth

If you are wondering what progress we are making in array notation ... basically nothing. We haven't even officially added a new entry at this point. But don't count ExE out yet. We also are really only getting started. All of this would be equivalent to a "second level tetrational array", that is using tetrational level arrays to describe the size of a tetrational array.

E100#^^#100#^#*###100

godteroogahlah-carta-tethrathoth

We won't go through all the googolisms again, although the carta-infix certainly makes that possible. Let's pick up the pace a little ...

E100#^^#100#^#*#^#100

deutero-godgahlah-carta-tethrathoth

This one might appear to be ambiguous. Does this mean deutero-"godgahlah-carta-tethrathoth" or "deutero-godgahlah"-carta-tethrathoth. Well the former doesn't make too much sense since "deutero" only works if we have a single delimiter that we can multiply by itself. However "godgahlah-carta-tethrathoth" actually has two delimiters: E100#^^#100#^#100. If we assume "deutero" simply doubles all the delimiters then we could disambiguate by calling E100#^^#*#^^#100#^#*#^#100 a deutero-godgahlah-carta-deutero-tethrathoth. Moving on ...

E100#^^#100#^#*#^#*#^#100

trito-godgahlah-carta-tethrathoth

A predictable continuation given the previous discussion. Again we can assume there is no ambiguity as long as we understand the multipliers can only apply to a single delimiter (the one it's attached to).

E100#^^#100#^##100

gridgahlah-carta-tethrathoth 

It's like we get a rogues gallery with all the highlights. At this point all that "ex-grand" stuff has been left unimaginably far in the dust, and yet we still are only just climbing out of the shadow of a tethrathoth.

E100#^^#100#^###100

kubikahlah-carta-tethrathoth

This is like a 3-dimensional array followed by a "2" in the next "tetrational block". We still are a long way to getting something we would consider a "larger array", but we will get there. ExE has plenty of room left to grow.

E100#^^#100#^#^#100

godgathor-carta-tethrathoth

Finally we find ourselves returning back to our starting point. What will happen when we get there? ...

E100#^^#100#^#^#^#100

godtothol-carta-tethrathoth

You get the idea. Let's move on to ...

E100#^^#100#^^#100 

tethratrithoth / tethrathoth-carta-tethrathoth

Finally we've mapped tethrathoth into itself. Here we can use our infix multiplier instead of the carta-infix. This would be equivalent to a tetrational array followed by 2 in the next tetrational-space ... it's a start ...

E100#^^#100#^^#100#^^#100

tethraterthoth / tethrathoth-carta-tethrathoth-carta-tethrathoth

This is an example of nesting the carta operator, showing that it can be stacked. More simply we can call this a tethraterthoth based on our infix multipliers.

E100#^^#*#5

tethrapethoth

Here we finally get our first new delimiter, '#^^#*#'. This technique however isn't really new. We have already (implicitly) defined hypernion multiplication in Cascading-E Notation (E^). Those same principles carry over to LECEN, xE^ and beyond. The rule is simple. If we have a delimiter of the form &*# then regardless of what & is @m&*#n = @m&m& ... &m w/n m's. Furthermore if we have hypernion G*H, and it's "decomposable, then (G*H)[n] = G*(H(n)). With that in mind we can go much much further, without inventing a single new rule. 

E100#^^#*#6

tethra-exthoth

Equivalent to a tetrational array followed by a 5 in the next tetrational space.

E100#^^#*#7

tethra-epthoth

Equivalent to a tetrational array followed by a 6 in the next tetrational space. 

E100#^^#*#8

tethra-octhoth

Equivalent to a tetrational array followed by a 7 in the next tetrational space.

E100#^^#*#100

tethrathoth-by-hyperion

Here we introduce a way to perform hypernion-multiplication on googolism's. G-by-H means that if G = E100A100, and H = E100B100 then G-by-H = E100A*B100. Pretty simple. By that logic, we could call this tethrathoth-by-grangol, but in this case I prefer to use "hyperion" to note that we are multiplying a "tethrathoth" (really #^^#) by a hyperion (#). This would be equivalent to a tetrational array followed by a 99 in the next tetrational space. At this point we've reached 1 additional entry past tetrational arrays ... but we are still only getting started, with both post-tetrational array spaces, and xE^.

E100#^^#*##100

tethrathoth-by-deutero-hyperion

Conveniently the meanings of deutero overlap here. Perhaps that's because I originally adapted them from proto-hyperions, deutero-hyperions, trito-hyperions, and so on. These terms came first as part of xE#. Since they cause repetitions of hyperions, it can be seen as a generalization that they cause repetitions of hypernions in general. Next up ...

E100#^^#*###100

tethrathoth-by-trito-hyperion

So it follows that #^^#*### corresponds to tethrathoth-by-trito-hyperion. Moving on ...

E100#^^#*#^#100

tethrathoth-by-godgahlah

This is the first example where we have used the "by" connector to combine two established googolisms. In terms of arrays, this represents the first row in the second tetrational space being filled. Although it's a common mantra that "arrays past tetrational arrays are ill-defined", this isn't precisely true. Such a space is most certainly well defined as long as tetrational space is well defined. In fact we can have an arbitrary number of tetrational spaces one after the other, without any issue. It's clear at this point that hypernion-multiplication is equivalent to "array-addition" at this point. Thus if we have arrays A and B represented by E100A100 and E100B100 respectively, then it follows that E100A*B100 will approximately yield a A+B array. Array Addition is well defined, since it is identical to ordinal addition.

E100#^^#*#^##100

tethrathoth-by-gridgahlah

This would be equivalent to the first plane of the second tetrational space.

E100#^^#*#^###100

tethrathoth-by-kubikahlah

This would be equivalent to the first cube (or "realm" in Bowers' verbage) of the second tetrational space.

E100#^^#*#^####100

tethrathoth-by-quarticahlah

This would be equivalent to the first tesseract (or "flune" in Bowers' verbage) of the second tetrational space.

E100#^^#*#^#^#100

tethrathoth-by-godgathor

This would be equivalent to the first dimensional block of the second tetrational space. That is, this is a X^^X+X^X array. We're starting to make progress, but we still are only getting started with LECEN. 

E100#^^#*#^#^#^#100

tethrathoth-by-godtothol

This would be equivalent to the first arbitrary-order-super-dimensional block of the second tetrational space. That is, this is a X^^X+X^X^X array. Which brings us to ...

E100#^^#*#^^#100

 deutero-tethrathoth

This would be equivalent to a pair of tetrational spaces in sequence. That is, it's an X^^X+X^^X array, or more compactly a (X^^X)*2 array. We've come a long way, but we still have far far to go. And the arrays keep coming. This is still as well defined as tetrational space, despite the common wisdom.

E100#^^#*#^^#*#^^#100 

trito-tethrathoth

Equivalent to a triplet of tetrational spaces. That is, a X^^X+X^^X+X^^X or (X^^X)*3 arrays.

E100#^^#*#^^#*#^^#*#^^#100

teterto-tethrathoth

Equivalent to a X^^X+X^^X+X^^X+X^^X or (X^^X)*4 arrays. Let's see where this leads ... 

E100(#^^#)^#100

hecato-tethrathoth

This would be equivalent to a row of a hundred tetrational spaces ... o_o

That is a (X^^X)*100 array. At this point we can see that a (#^^#)^# delimiter is capable of creating (X^^X)*X arrays of arbitrary size.

On the ExE side, this is the first example we have of an expression of the form a^# where a != #. Actually it was possible to do this long before xE^. E^ was the first introduction of Hypernion-exponentiation, and the rule applies equally well to ##^# as to #^#. In all cases a^#[n] is just equivalent to a*a*...*a w/n a's. 

E100(#^^#)^#100#2

grand hecato-tethrathoth

Here we would have a hecato-tethrathoth tetrational spaces in a row! Ikes! Still, this is just what happens when we start on "second level" row-of-tetrational-space arrays. Neither ExE nor BEAF is anywhere near to out, so let's keep going ...

E100(#^^#)^#100(#^^#)^#100

tethratrifact

This one is a little unusual but it's based on using the name "tethrafact" for a "hecato-tethrathoth". In array notation this would be a row of tetrational spaces, followed by a 2 in the next row of tetrational spaces. Seems like a pretty small improvement now, but remember we had to skip all ExE expressions of the form E100(#^^#)^#100&100 where & is a delimiter of lower rank than (#^^#)^#, or in fact any row of delimiters whose ranks are all lower than (#^^#)^#.

E100(#^^#)^#*(#^^#)^#100

deutero-tethrafact

So, keeping track of our progress, this would be two full rows of tetrational spaces. That would be equivalent to (X^^X)*X2. Wild.

E100(#^^#)^##100

tethrathothigrid

Here is another more modern name for this number. This would be equivalent to a 100x100 plane of tetrational spaces. We've reached (X^^X)*X^2 arrays. Yes, this is still a totally valid and easily understood array, and we are still only getting started with both BEAF and ExE.

E100(#^^#)^###100

kubicutethrathoth / tethrathothicube

This number actually has two canonical names. Following the same patterns as before this would be a cube of tetrational spaces, or a size 100 (X^^X)*X^3 array. Nifty.  

E100(#^^#)^####100

quarticutethrathoth / tethrathothitess

A tesseract of tetrational spaces or a size 100 (X^^X)*X^4 array.

E100(#^^#)^#^#100

centicutethrathoth / tethragodgathor

Things are starting to get weird. This is a dimensional array of tetrational spaces. That is, a (X^^X)*X^X array. Still nothing even close to stopping either notation from continuing.

E100(#^^#)^#^#^#100

tethragodtothol

This would be equivalent to an abritrary super-dimensional space composed of tetrational spaces. A (X^^X)*X^X^X array. Array Notation is still well defined.

E100(#^^#)^#^#^#^#100

tethragodtertol

This would be equivalent to an abitrary trimensional array composed of tetrational spaces. A (X^^X)*X^X^X^X array. So what does this eventually lead to? 

E100(#^^#)^(#^^#)100

tethra-tethrathoth / tethraduliath

Here we at last arrive at a tetrational array made up of tetrational arrays. That is, this is a (X^^X)*(X^^X) or (X^^X)^2 array. The ability to take a structure and duplicate it in this way is something I like to call "shelling". Technically dimensional space is just the "shelling" of linear space. In the same way we can take a tetrational space and shell it to create a new kind of space. Let's call this a tetration-squared space.

E100(#^^#)^(#^^#*#)100

tethradulifact

The names are starting to get a little weird, but keep in mind there are a loooot of names, thousands of them, and we are skipping most of them here. In any case this would be equivalent of a row of tetration-squared spaces, a (X^^X)^2*X array. I think you can see where this is going.

E100(#^^#)^(#^^#*##)100

tethraduligrid

So here we have a plane of tetration spaces of tetration spaces, a (X^^X)^2*X^2. Obviously we can keep going like this ...

E100(#^^#)^(#^^#*#^#)100

tethraduli-godgathor

This name might require some explanation. This could be treated as equivalent to E100((#^^#)^(#^^#))^#^#100, which is a^#^# where a = '(#^^#)^(#^^#)'. Thus this can be thought of as a strong modification of a godgathor. This would be, following the pattern as before, a ... dimensional space of tetrational spaces of tetrational spaces. At this point we are at (X^^X)^2*X^X arrays ...  

E100(#^^#)^(#^^#*#^^#)100

tethrathruliath

This is a (X^^X)^3 array, a tetrational array of tetrational arrays of tetrational arrays. It might seem a little weird that we have (#^^#)^(#^^#)^2 to represent (X^^X)^3, but this makes more sense when you realize that there is an offset between hyperions and Xs. We can think of (#^^#)^(#^^#*#^^#) as of equivalent rank to #^(#^^#*#^^#*#^^#) = #^(#^^#)^3. Dropping the # base we obtain the equivalent X structure. At this point it should be clear that we can nest or "shell" the tetrational space arbirarily many times. Bowers' doesn't have any specific googolisms for this level, but we have a nice milestone coming up in ExE. But first ...

E100(#^^#)^(#^^#*#^^#*#^^#)100

tethrateruliath

This is a (X^^X)^4 array. Now let's go to the general case ... 

E100(#^^#)^(#^^#)^#100

Monster-Giant

Here is a fun number. This expands to:

E100(#^^#)^(#^^#*#^^#* ... ... ... ... *#^^#*#^^# w/100 #^^#'s)100

This would be a (X^^X)^101 array, a size 101 (X^^X)^X array. Now we can imagine nesting tetrational spaces an arbitrary number of times which brings us to ... 

E100(#^^#)^(#^^#)^#100#2

Grand Monster-Giant

This would be a (X^^X)^(Monster-Giant+1) array. Where could we possibly go from here? Seems like we've reached the end of what makes sense in array notation, right? Wrong ...

E100(#^^#)^((#^^#)^#*#^^#)100

tethra-Monster-Giant

Alright, what in the hell is this?! Well we can keep going. Note that this is much much much much larger than a mere E100(#^^#)^(#^^#)^#101. To explain let's call a (X^^X)^X a shelled-tetrational space. So this number is a tetrational space composed of shelled-tetrational spaces. Woah. So basically this is a (X^^X)^X*(X^^X) or (X^^X)^(X+1) array. Just like when we reach infinite dimensional space we can keep going with the X+1 dimension, so with tetrational space we can continue with the X+1 order "tetrational space". We've kind of run out of language to describe it, but the mathematics should be clear. It's clear that we can continue in this way. Let's look at a few examples ...

E100(#^^#)^(#^^#)^##100

Monster-Grid

Here is a cool extension. This number would be like a 2-super-dimensional-tetrational array, that is, it would be equivalent to a (X^^X)^X^2 array. We could also call this a X^2 order tetrational array. Think of shelling a tetrational structure and obtaining a new structure. So what happens when we shell that new structure? We get this. Every time we shell the previous structure we actually get the next "super dimension". This doesn't magically stop at tetrational arrays, but keeps going even after it. So this would be two super-dimensions past tetrational arrays. Imagining and making sense of this is a little tricky, however I contend that this is still perfectly well defined and presents no real issue. If we compute the number of entries of such a structure it will be exactly (p^^p)^p^2, where p is the value of the prime entry. We could keep going but things get murky at some point before we reach ...

E100(#^^#)^(#^^#)^(#^^#)^#100

Super Monster-Giant

In ExE this number is perfectly well defined, but in Array Notation this is where we hit something of a wall, with unclear array structures. If the climbing method is true however jack has only just begun his climb up to the top of the beanstalk and we are nowhere near what you would get if you actually computed (p^^p)^(p^^p)^p which would instead be approximately p^^(p+1). It is based on this theory that array notation was believed to reach X^^(X2) arrays by e(1). Speaking of which ... 

E100(#^^#)^^#100

terrible tethrathoth

Here we have the first ExE googolism to reach the level of e(1). This is still relatively tame as far epsilon numbers go. We've got a long way to go before we would require a new ordinal function. In terms of arrays this is believed by some to be a X^^(X2) array. 

E100(#^^#)^^#*#100

terrible-tethrathoth-by-hyperion

Take whatever the previous array structure was for terrible tethrathoth, and have a 100 in the very next entry, and you would have this number.

E100(#^^#)^^#*##100

terrible-tethrathoth-by-deutero-hyperion

Place 100 in the second entry after an e(1)-array (whatever that means).

E100(#^^#)^^#*###100

terrible-tethrathoth-by-trito-hyperion

Place 100 in the third entry after an e(1)-array.

E100(#^^#)^^#*####100

terrible-tethrathoth-by-teterto-hyperion

Place 100 in the fourth entry after an e(1)-array.

E100(#^^#)^^#*#^#100

terrible-tethrathoth-by-godgahlah

The first row after an e(1)-array.

E100(#^^#)^^#*#^#*#^#100

terrible-tethrathoth-by-deutero-godgahlah

The first two rows after an e(1)-array.

E100(#^^#)^^#*#^^#100

terrible-tethrathoth-by-tethrathoth

The first tetrational array after an e(1)-array.

E100(#^^#)^^#*(#^^#)^^#100

deutero terrible-tethrathoth

a row of two e(1)-arrays.

E100(#^^#)^^#*(#^^#)^^#*(#^^#)^^#100

trito terrible-tethrathoth

A row of three e(1)-arrays. 

E100((#^^#)^^#)^#100

hecato-terrible tethrathoth

A row of 100 e(1)-arrays. alternatively ((#^^#)^^#)^# is equivalent to an arbitrary sized row of e(1)-arrays.

E100((#^^#)^^#)^##100

territethrathothigrid

A plane of e(1)-arrays.

E100((#^^#)^^#)^###100

terribtethrathothicube

A cube of e(1)-arrays.

E100((#^^#)^^#)^#^#100

terrible-tethrathoth-ipso-godgahlah

An arbitrary dimensional array of e(1)-arrays.

E100((#^^#)^^#)^#^#^#100

terrible-tethrathoth-ipso-godgathor

An arbitrary super-dimensional array of e(1)-arrays

E100((#^^#)^^#)^(#^^#)100

terrible-tethrathoth-ipso-tethrathoth

A tetrational array of e(1)-arrays. Cray cray.

E100((#^^#)^^#)^(#^^#)^(#^^#)100

terrible-tethrathoth-ipso-tethrathoth-ipso-tethrathoth

A tetration order tetration array of e(1)-arrays ... o_o;

E100((#^^#)^^#)^(#^^#)^(#^^#)^(#^^#)100

terrible-tethrathoth-ipso-tethrathoth-trebletetrate

A ... uh ... tetration order tetration order tetration array of e(1)-arrays. The names are getting pretty ridiculous but again keep in mind how many thousands of names were needed. The names are actually pretty literal and straight forward. "tethrathoth-trebletetrate" basically just means #^^# tetrated to the third. If this thing sounds like some kind of molecule, well, that was sorta the point. I wanted the names to start getting super complicated sounding like this.

E100((#^^#)^^#)^((#^^#)^^#)100

dubletetrated-terrible-tethrathoth

A e(1)-array made up of of e(1)-arrays. That is, a 2nd order e(1)-array. Name should be self explanatory. Basically it's saying this is E100((#^^#)^^#)^^(2)100.

E100((#^^#)^^#)^((#^^#)^^#)^((#^^#)^^#)100

trebletetrated-terrible-tethrathoth

A 2nd order e(1)-array of e(1)-arrays.

E100((#^^#)^^#)^((#^^#)^^#)^((#^^#)^^#)^((#^^#)^^#)100

quadrupletetrated-terrible-tethrathoth

Er ... a 2nd order e(1)-array order e(1)-array of e(1) arrays :| 

E100((#^^#)^^#)^^#100

terrible terrible tethrathoth

This is on the order of e(2). Following the conventional wisdom this would be equivalent to a X^^(X3) array.

E100(((#^^#)^^#)^^#)^^#100

terrible terrible terrible tethrathoth

This is on the order of e(3). Following the conventional wisdom this would be equivalent to a X^^(X4) array. 

E100((((#^^#)^^#)^^#)^^#)^^#100

terrible terrible terrible terrible tethrathoth

This is on the order of e(4). Following the conventional wisdom this would be equivalent to a X^^(X5) array.

E100#^^#>#100

tethriterator / tethrathoth ba'al / ninety-nine-ex-terrible tethrathoth

This number begins to push the limits of LECEN. In xE^ however this can be compactly expressed as E100#^^#>#100, using the new #^^#># delimiter which diagonalizes over LECEN with the fundamental sequence:

#^^#>#[1] = #^^#

#^^#>#[n] = (#^^#>#[n-1])^^#

With this in place it now becomes trivial to move way way beyond LECEN...

In terms of ordinals, this would be at the level of e(w). We still are in the "epsilon epoch" but we are just beginning to get somewhere within it. This is believed to be equal to a X^^X^2 array, although it's only a X^^(X+1) array according to the theory of the climbing method.

E100#^^#>#(1+E100)

googol-ex-terrible tethrathoth

Let the climb begin. This would be E100(...((#^^#)^^#)...)^^#100 with a googol+1 ^^#'s. 

E100#^^#>#(1+E100#100)

grangol-ex-terrible tethrathoth

Alright let's get through all the highlights. This is a "size grangol e(w)-array" ... 

E100#^^#>#(1+E100#100#100)

greagol-ex-terrible tethrathoth

A size greagol e(w)-array ...

E100#^^#>#(1+E100#100#100#100)

gigangol-ex-terrible tethrathoth

A size gigangol e(w)-array ... 

E100#^^#>#(1+E100##5)

gorgegol-ex-terrible tethrathoth

A size gorgegol e(w)-array ...  

E100#^^#>#(1+E100##6)

 gulgol-ex-terrible tethrathoth

A size gulgol e(w)-array ... 

E100#^^#>#(1+E100##7)

gaspgol-ex-terrible tethrathoth

A size gaspgol e(w)-array ... 

E100#^^#>#(1+E100##8)

 ginorgol-ex-terrible tethrathoth

A size ginorgol e(w)-array ...

E100#^^#>#(1+E100##100)

gugold-ex-terrible tethrathoth

A size gugold e(w)-array ...

E100#^^#>#(1+E100##100##100)

gugolthra-ex-terrible tethrathoth

A size gugolthra e(w)-array ...

E100#^^#>#(1+E100###100)

throogol-ex-terrible tethrathoth

A size throogol e(w)-array ...

E100#^^#>#(1+E100####100)

teroogol-ex-terrible tethrathoth

A size teroogol e(w)-array ...

E100#^^#>#(1+E100#^#100)

godgahlah-ex-terrible tethrathoth

A size godgahlah e(w)-array ...

E100#^^#>#(1+E100#^##100)

gridgahlah-ex-terrible tethrathoth

A size gridgahlah e(w)-array ...

E100#^^#>#(1+E100#^###100) 

kubikahlah-ex-terrible tethrathoth

A size kubikahlah e(w)-array ...

E100#^^#>#(1+E100#^####100) 

quarticahlah-ex-terrible tethrathoth

A size quarticahlah e(w)-array ...

E100#^^#>#(1+E100#^#^(5)100)

quinticahlah-ex-terrible tethrathoth

A size quinticahlah e(w)-array ... 

E100#^^#>#(1+E100#^#^(6)100)

sexticahlah-ex-terrible tethrathoth

A size sexticahlah e(w)-array ...

E100#^^#>#(1+E100#^#^(7)100)

septicahlah-ex-terrible tethrathoth

A size septicahlah e(w)-array ...

E100#^^#>#(1+E100#^#^(8)100)

octicahlah-ex-terrible tethrathoth

A size octicahlah e(w)-array ...

E100#^^#>#(1+E100#^#^(9)100)

nonicahlah-ex-terrible tethrathoth

A size nonicahlah e(w)-array ...

E100#^^#>#(1+E100#^#^(10)100)

decicahlah-ex-terrible tethrathoth

A size decicahlah e(w)-array ... 

E100#^^#>#(1+E100#^#^(20)100)

viginticahlah-ex-terrible tethrathoth

A size viginticahlah e(w)-array ...

E100#^^#>#(1+E100#^#^(90)100)

nonaginticahlah-ex-terrible tethrathoth

A size nonaginticahlah e(w)-array ... 

E100#^^#>#(1+E100#^#^#100)

godgathor-ex-terrible tethrathoth

A size godgathor e(w)-array ...

E100#^^#>#(1+E100#^#^#101)

goober-bunch-ex-terrible tethrathoth

A size goober bunch e(w)-array. 'member goober bunch ... even Pepperidge Farm doesn't remember any more, that was a couple epochs ago now!

E100#^^#>#(1+E100#^#^##100)

gralgathor-ex-terrible tethrathoth

A size gralgathor e(w)-array ... 

E100#^^#>#(1+E100#^#^###100)

thraelgathor-ex-terrible tethrathoth

A size thraelgathor e(w)-array ... 

E100#^^#>#(1+E100#^#^####100)

terinngathor-ex-terrible tethrathoth

A size terinngathor e(w)-array ... 

E100#^^#>#(1+E100#^#^#^#100)

godtothol-ex-terrible tethrathoth

A size godtothol e(w)-array ... 

E100#^^#>#(1+E100#^#^#^##100)

graltothol-ex-terrible tethrathoth

A size graltothol e(w)-array ... 

E100#^^#>#(1+E100#^#^#^###100)

thraeltothol-ex-terrible tethrathoth

A size thraeltothol e(w)-array ... 

E100#^^#>#(1+E100#^#^#^####100)

terinntothol-ex-terrible tethrathoth

A size terinntothol e(w)-array ... 

E100#^^#>#(1+E100#^#^#^#^#100)

godtertol-ex-terrible tethrathoth

A size godtertol e(w)-array ... 

E100#^^#>#(1+E100#^#^#^#^##100)

graltertol-ex-terrible tethrathoth

A size graltertol e(w)-array ... 

E100#^^#>#(1+E100#^^(6)100)

godtopol-ex-terrible tethrathoth

A size godtopol e(w)-array ...

E100#^^#>#(1+E100#^^(7)100)

godhathor-ex-terrible tethrathoth

A size godhathor e(w)-array ...

E100#^^#>#(1+E100#^^(8)100)

godheptol-ex-terrible tethrathoth

A size godheptol e(w)-array ... 

E100#^^#>#(1+E100#^^(9)100)

godoctol-ex-terrible tethrathoth

A size godoctol e(w)-array ... 

E100#^^#>#(1+E100#^^#100)

tethrathoth-ex-terrible tethrathoth

A size tethrathoth e(w)-array ... 

E100#^^#>#(1+E100#^^#100#2)

grand-tethrathoth-ex-terrible tethrathoth

A size grand tethrathoth e(w)-array ...

E100#^^#>#(1+E100(#^^#)^(#^^#)^#100)

Monster-Giant-ex-terrible tethrathoth

A size Monster-Giant e(w)-array ...

E100#^^#>#(1+E100(#^^#)^(#^^#)^(#^^#)^#100)

Super-Monster-Giant-ex-terrible tethrathoth

A size Super Monster-Giant e(w)-array ...

E100#^^#>#(1+E100(#^^#)^^#100)

terrible-tethrathoth-ex-terrible tethrathoth

A size terrible tethrathoth e(w)-array ... 

E100#^^#>#(1+E100((#^^#)^^#)^^#100)

terrible-terrible-tethrathoth-ex-terrible tethrathoth

A size terrible terrible tethrathoth e(w)-array ...  

E100#^^#>#(1+E100#^^#>#4)

terrible-terrible-terrible-tethrathoth-ex-terrible tethrathoth

A size terrible terrible terrible tethrathoth e(w)-array ... 

E100#^^#>#(1+E100#^^#>#5)

terrible-terrible-terrible-terrible-tethrathoth-ex-terrible tethrathoth

A size terrible terrible terrible terrible tethrathoth e(w)-array ... 

E100#^^#>#100#2

= E100#^^#>#(E100#^^#>#100)

= E100((...((#^^#)^^#)...)^^#)^^#100 w/tethrathoth ba'al "^^#"s

The Great and Terrible Tethrathoth / grand tethriterator

At the end of the original run of the introduction of Cascading-E Notation I ended with this number, which I called a "The Great and Terrible Tethrathoth". With the introduction of Extended Cascading-E Notation, it was eventually renamed grand tethriterator. This number basically is about at the limit of what we can reasonably expect to express in LECEN, which is an acronym for "Limited Extension Cascading-E Notation". This was prior to the invention of the "caret-top" which is the symbol (>). Some might wonder why I stopped at this point during the introduction fo Cascading-E Notation. The problem was I didn't have a neat way to notate what arbitary applications of the tetrational operator ^^# were in terms of the climbing method. According to Bowers this would be exactly equivalent to a X^^(X+1) array. Without the invention of the caret-top the only reasonable next expression seemed to be #^^##, but I knew that wouldn't happen until much later, so I needed something to fill the gap between ((...((#^^#)^^#)...)^^#)^^# and #^^##. Some googologists at the time actually went ahead and set them equal to each other. Worse there was a good justification for this via the Knuth Arrow theorem, where we would add the tetrates, implying that we would get #^^(#+#+..+#) = #^^## (even though this is actually only approximate). I didn't want ExE to fall behind the climbing method which is why the caret-top was invented. It was partially inspired by Bowers own notation for post-tetrational arrays in which {X^^X}^{X} was not interpretted as (X^^X)^X but rather as if there was a whole stack of X's and another X on top. This eventually became my own notation #^^#>#. The caret-top acts as an "iterator" (hence the name tethriterator). It iterates whichever operation it is attached to applied ordinally many times to the first argument. Thus A@B>C will form a ternary operation that extends whichever binary operation, @, it is applied to. This fills out all the gaps and allows us to create Extended-Cascading-E Notation. There is just one problem. No one has been able to show where (X^^X)^X actually fall along order-types. And this is why I've been unable to determine how much further one would have to go to reach 3^^^3&3. That said let's go a little further with these constructs before going over to the next epoch ...

E100#^^#>#(1+E100#^^#>#(1+E100#^^#100))

tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth

We can repeat this construct and be quite sure that we will be nowhere near 3^^^3&3. Why? Because at very least 3^^^3&3 would be approximately equal to 3^3#^^##3, which is 3^3#^^#>#^^#>#^^#3. This goes way way beyond what we are doing here. Now let's nest this sucker into oblivion ... well not literally ... that comes much much later ... 

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

tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth

            That's a hundred tethrathoth's in all, and yet this massive construction would be smaller than  E100#^^#>#100#100  which would be massively smaller than  E100#^^#>#100#^^#>#100 which would be massively smaller than E100#^^#>##100 which would be massively smaller than E100#^^#>#^^#100 which wouldn't come even close to  3^^^3&3 even in it's most modest form.  Extended Cascading-E Notation (xE^) has been developed far far beyond this, and even at it's smallest we are still a VERY VERY VERY VERY ... VERY VERY LONG WAY to pentational arrays. In the next epoch are some unimaginably vast numbers which currently no one quite knows where they should fall on the Fast-growing hierarchy ...