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UFNL2


ULTIMATE FINITE NUMBERS LIST
PART II

numbers expressible with functions between order-types w and w^w



Conway-Arrow Epoch

E100##100 ~ E100###100
Entries: 54


E100##1#2

great googol

                    The great googol is E100#100#100#100# ... #100#100#100 with a googol 100s. This can be written compactly as E100##1#2 by utilizing the power of the deutero-hyperion. This number is much larger than a gugold but much smaller than a Moser. It can be approximated as 10^^^...^^^100 in which there are a googol up-arrows.

       



2[2[5]]

Moser

                This number is 2 inside a "mega-gon" (See mega). This number was popularly regarded as "the largest number", until Graham's Number came along. A Moser is less than G(2), but larger than G(1). It's approximately:

2<10^^257>3


G(2)

3<3<4>3>3

 


g(2)

2<2<12>3>3 

 



E100##100#2

gugolda-suplex

 


G(3)

3<3<3<4>3>3>3

 


g(3)

2<2<2<12>3>3>3



E100##100#3


gugolda-dusuplex

 


G(4)

3<3<3<3<4>3>3>3>3

 


g(4)

2<2<2<2<12>3>3>3>3

 


g(7)

Little Graham

                This is the original Graham's Number, which first appeared in RL Graham's 1971 paper "Ramsey Theory for n-parameter sets". This is currently the smallest known upper-bound for the problem in Ramsey theory for which "Graham's Number" as we know it today, was supposedly invented. It is significantly smaller than Graham's Number, but still much much bigger than a Moser. Little Graham can be defined most compactly as follows:

Little Graham = g(7)

where g(1) = 2^^^...^^^3 w/12 ^s

and g(n) = 2^^^...^^^3 w/g(n-1) ^s whenever n>1

Using Jonathan Bowers' hyper-operator notation we have:

Little Graham = 2<2<2<2<2<2<2<12>3>3>3>3>3>3>3 <3,9,1,2> << <10,100,1,2>


G(8)

This is the smallest member of the Graham Series, G(n), greater than Little Graham. Consequently:

Little Graham = g(7) < G(8) << G(64) = Graham's Number

:: Little Graham << Graham's Number

 


G(64)

Graham's Number

            Graham's Number, sometimes abbreviated as G, was first coined in 1977 when Ronald Graham used it as an upper bound to a problem in Ramsey theory. At the time it was the largest number anyone had seen arising from a serious mathematical proof. It quickly gained recognition in the Guinness Book of World Records for being the largest number in serious mathematics. But what is Graham's Number,...(READ MORE)


 
A(G64,G64)
 
xkcd
 

G(65)

G of sixty-five

            A common retort to Graham's Number.



G(100)

G of one hundred

            Nice try. This is still technically smaller than a "graatagold", since a graatagold is the 100th member of the gugold series, and G(1) < gugold, G(2) < gugolda-suplex, G(3) < gugolda-dusuplex etc. Therefore: G(100) < graatagold.



E100##100#100

graatagold


This is the smallest of my googolism's to exceed Graham's Number. It is comparable to Jonathan Bowers'Corporal = <10,100,1,2>.



g(10^10^102,1,1,2,100,10)

Joyce's googolplux

Joyce's mysterious googolplux. This number is defined using Joyce's 6-argument g-function and it's much larger thanGraham's Number. It is approximately G(10^10^102). 




G(3^^^^3)

G of three hexated to the three

Yudkowsky's Number

            A slightly better retort to Graham's Number (See G(65)). I call this number Yudkowsky's Number. Elizier Yudkowsky mentioned it in his article "Staring at the singularity", to make the point that Graham's Number is probably not even the largest number at this particular complexity level.

            Suffice it to say that this is rather underwhelming from a googologist's point of view. It's less than a single recursive step from Graham's Number.

            Another form of this number is G(G(1)), which begs the question ... why stop there!



g(g(8,10^100,10^100),1,1,2,100,10)

Joyce's googolpluc

Here's a number to make a "singularities head spin", Joyce's googolpluc. It is approximately G(googol^^^^^^^googol).



g(g(7625597484986,10^100,10^100),1,1,2,100,10)

Joyce's googolplum

Joyce's googolplum is approximately G(googol^^^...^^^googol) w/7,625,597,484,985 ^s. Even so this number is still vastly smaller than the next entry ...


G(G(26))

                    A lower bound on Conway's tetratri in terms of Graham's function.


3-->3-->3-->3

Conway's tetratri

                    A number mentioned by Jonathan Bower's as the largest number in the professional literature, and referred to as Conway's three-three-three-three. This number first appeared in Conway's Book of Numbers (1996) as an example of a number larger than Graham's Number. It has since been surpassed by even larger numbers in the professional mathematical literature.

                    This is also the 19th record setting entry in the "My Number is Bigger" competition, and the 21st entry. It was entered by Gmalivuk after giving up on a modest response on the previous record holder and he just decided to jump to the introduction of Conway Chain-Arrows. From this point onwards, up-arrow notation will no longer be competitive! This is also the record holder by the end of the first page of the forum ... of which there are 35 pages!!!


G(G(27))

                    An upper bound on Conway's tetratri in terms of Graham's function.


G(G)

G of G

            A slightly less naive response to Graham's Number (see G(3^^^^3)). This number can also be written G(G(64)). This number however is still less than G(G(100)), which is still less than a graatagolda-sudex!



G(G(100))



E100##100#100#2

graatagolda-sudex



E100##100##3

greegold



4-->4-->4-->4

Conway's tetratet

                    This is the largest explicitly defined finite number in Conway's Book of Numbers. It's also the 4th Conway Number.


E100##100##4

grinningold



E100##100##5

golaagold



E100##100##6

gruelohgold



E100##100##7

gaspgold



E100##100##8

ginorgold



E100##100##100

gugolthra



E100##100##100##2

graatagolthra



E100##100##100##3

greegolthra



E100##100##100##4

grinningolthra



E100##100##100##5

golaagolthra



E100##100##100##6

gruelohgolthra



E100##100##100##7

gaspgolthra



E100##100##100##8

ginorgolthra



E100##100##100##100

gugoltesla



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

graatagoltesla



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

greegoltesla



E100##100##100##100##4

grinningoltesla



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

golaagoltesla



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

gruelohgoltesla



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

gaspgoltesla



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

ginorgoltesla



E100##100##100##100##100

gugolpeta



E100##100##100##100##100##100

gugolhexa



E100##100##100##100##100##100##100

gugolhepta



E100##100##100##100##100##100##100##100

gugolocta


Linear Array Epoch

E100###100 ~ E100#^#100
Entries: 41


E100###100

throogol



E100###100#100

thrangol



cgG64(G64)

Mouffles Monster

                    This is the 20th record setting entry in the "My Number is Bigger" competition. It is also the first record setter on page 2. It was entered by Mouffles in response to Gmalivuk's chain-arrow entry. In order to describe it we need a new function. Let...

cg(1) = 1
cg(2) = 2-->2
cg(3) = 3-->3-->3
and in general
cg(n) = n-->n-->n--> ... ... -->n-->n-->n w/n n's

Mouffles Monster = cgGraham's Number(Graham's Number)

                    Note that Mouffles has simply created the "cg" function first described by Conway and Guy in the Book of Numbers. At this point chain-arrows are no longer competitive. Despite the terrific leap forward, in the larger recursive scheme of things this nesting of cg(n) only leads to an increase of an order-type or two from Chain-arrows which is already of order-type w^2. This means nesting cg(n) falls around w^2+2. I've estimated that it falls between a thrangol and a threagol. At this point all the popular large number notations are exhausted, so the contestants are forced to either start inventing notations or pull something more esoteric from the professional literature such as FGH.


 
E100###100#100#100

threagol



E100###100#100#100#100

thrigangol



E100###100#100#100#100#100

throrgegol



E100###100#100#100#100#100#100

thrulgol



E100###100#100#100#100#100#100#100

thraspgol



E100###100#100#100#100#100#100#100#100

thrinorgol



E100###100##100

thrugold



E100###100##100##2

thraatagold



E100###100##100##3

threegold



E100###100##100##4

thrinningold



E100###100##100##5

throlaagold



E100###100##100##6

thruelohgold



E100###100##100##7

thraspgold



E100###100##100##8

thrinorgold



E100###100##100##100

thrugolthra



E100###100##100##100##100

thrugoltesla



E100###100##100##100##100##100

thrugolpeta



E100###100##100##100##100##100##100

thrugolhexa



E100###100##100##100##100##100##100##100

thrugolhepta



E100###100##100##100##100##100##100##100##100

thrugolocta



E100###100###100

tristo-throogol



E100###100###100###100

teristo-throogol



E100###100###100###100###100

pesto-throogol



E100###100###100###100###100###100

existo-throogol



E100###100###100###100###100###100###100

episto-throogol



E100###100###100###100###100###100###100###100

ogisto-throogol



E100####100

teroogol



E100####100###100

teri-throogol



E100####100####100

tristo-teroogol



E100####100####100####100

teristo-teroogol



E100####100####100####100####100

pesto-teroogol



E100####100####100####100####100####100

existo-teroogol



E100####100####100####100####100####100####100

episto-teroogol



E100####100####100####100####100####100####100####100

ogisto-teroogol



E100#####100

petoogol



E100######100

ectoogol



E100#######100

zettoogol



E100########100

yottoogol


Dimensional Array Epoch
E100#^#100 ~ E100#^#^#100
Entries: 279


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

godgahlah

                This number pushes the limit of xE#, and soon we will be beyond what we can practically expression even in xE#. However this is just the beginning of Cascading-E Notation. In Cascading-E we can write a godgahlahmore simply as E100#^#100.
 


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

godgahlahgong

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



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

grand godgahlah

            Also can be written E100#^#100#2 = E100#^#(E100#^#100) = E100#^#godgahlah



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

grand godgahlahgong

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



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#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 competitorIn 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 grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand grand 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 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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 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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 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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 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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 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.

                    Continue on to part 3 for numbers so large they can't even be expressed in linear array notation ...