ULNL2

Back to Ultimate Large Numbers List Part I

ULTIMATE LARGE NUMBERS LIST

PART II

Hyper-E & Extended Hyper-E

Entries: 405

Numbers larger than those expressible with E-Notation but still expressible in E# and xE#.

10^10^10^10^10^10^10^10^10^10

dekalogue / decker

10^^10 = E1#1#2 = E1#10 = E10#9

This is the 6th largest entry on Robert Munafo's Number List. That really says something about how the list skews towards the small power towers and barely has any of the larger power towers in the range of the entries. The entries grow really fast from here jumping from 10^^10, to approximately 10^^24 to 10^^163 to 10^^257 to 10^^997 and then suddenly a jump all the way to 10^^(10^308). Our climb here will be a bit more gradual.

This is one of Jonathan Bowers' original googolism's to appear on his list of Infinity Scrapers. He calls this number decker. I use the name dekalogue instead, and it also leads to a series of other extended names by changing the suffix.

The dekalogue may be expressed in many different ways. In Donald Knuth's Up-Arrow notation we may denote it most succinctly as 10^^10. In Bowers' notations it may be denoted as 10<2>10 or <10,10,2>. In Hyper-E it can be denoted by E1#10 or E1#1#2 since E1#1#2 = E1#(E1#1#1), and E1#1#1 = E1#1 = E1 = 10^1 = 10. E1#1#2 is significant in that it's the smallest possible non-trivial 3-argument Hyper-E Expression. The smallest possible 3-argument Hyper-E expression would be E1#1#1 however this is trivial since it can be reduced to a single argument of E1. See E2#1#2.

10^10^10^10^10^10^10^10^10^100

E100#9

googoloctiplex

The 9th member of the googol sequence.

10^10^10^10^10^10^10^10^10^10^10

10^^11 = E1#11

endekalogue

At this point writing out power towers is becoming increasingly impractical. Here we have two optional short hands. For b^b^...^b w/p b's we can either notate this using knuth arrows as b^^p or we can use generalized Hyper-E notation as b^1#p. If b=10 we can also write it as E1#p. We can continue the logue sequence past the 10th member by continuing to use greek, same as is used in the polygon names.

10^10^10^10^10^10^10^10^10^10^100

E100#10

googolnoniplex

The 10th member of the googol sequence. Note that the nth member of the googol series will always have n 10s topped by 100.

10^10^10^10^10^10^10^10^10^10^10^10

10^^12 = E1#12

dodekalogue

Similar to dodecahedron we have dodekalogue for the 12th of the logue series.

10^10^10^10^10^10^10^10^10^10^10^100

E100#11

googoldeciplex

The 11th member of the googol sequence.

10^10^10^10^10^10^10^10^10^10^10^10^10

E1#13

triadekalogue

The 13th of the logue sequence. 

13^13^13^13^13^13^13^13^13^13^13^13^13

13^^13

megafugathirteen

                    This number was the 11th valid entry in the "My Number is Bigger" competition shortly after Rodan's 10^10^10^10. This number was entered by Blatm in the form D^^D where "D" was hexadecimal for 13.

10^10^10^10^10^10^10^10^10^10^10^10^10^10

E1#14

tetradekalogue

The 14th official name in the logue sequence.

10^10^10^10^10^10^10^10^10^10^10^10^10^10^10

E1#15

pentadekalogue

The 15th official name in the logue sequence.

10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10

E1#16

hexadekalogue

The 16th official name in the logue sequence.

10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10

E1#17

heptadekalogue

The 17th official name in the logue sequence.

10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10

E1#18

octadekalogue

The 18th official name in the logue sequence.

10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10

E1#19

ennadekalogue

The 19th official name in the logue sequence.

10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10

10^^20 = E1#20

icosalogue / giggup

The icosalogue is based on the icosahedron. In both cases the suffix icosa means 20. At this point we run into a little bit of a problem, as there is not a nice way to turn 21 in greek into a suffix. This is because the order of terms switches from ones-tens to tens-ones. 21 would be approximately "icosi ena" but there doesn't seem to be a good way to adapt this into a suffix. As a result the googolism's for the logue series begin to jump by 10s. It's also worth noting that this particular number is to the guppi what the giggol is to the googol. googol = 10^100 --> giggol = 10^^100. guppi = 10^20 --> 10^^20. Following this concept I have coined the giggup for this number, a portmanteau of giggol + guppi. 

20^20^20^20^20^20^20^20^20^20^20^20^20^20^20^20^20^20^20^20

20^^20

megafugatwenty

The megafugatwenty would be a number slightly larger than a giggup. megafuga is a suffix created by Alistair Cockburn for n^^n.

10^^30 = E1#30

triantalogue

This is the next named member of the logue series after icosalogue.

10^^40 = E1#40

terantalogue

The 40th member of the logue sequence. Based on saranta, 40 in greek. Changed to teranta to make it more closely resemble tetra, 4 in greek.

E1#50

penantalogue

The 50th member of the logue sequence.

E1#60

exatalogue

The 60th member of the logue sequence.

E1#70

eptatalogue

The 70th member of the logue sequence.

E1#80

ogdatalogue

The 80th member of the logue sequence.

E1#90

entatalogue

The 90th member of the logue sequence.

10^^100 = E1#100

hectalogue / giggol

            The giggol is the first of Jonathan Bower's original extensions to the googol series. It is also the 4th largest number to appear on Robert Munafo's Number list, so we could say the very large numbers begin here. The giggol can be written in Hyper-E Notation as E1#100.

(10^^100)^2

gargiggol

            This might seem like a significant improvement over the giggol, like having a googolplex googolplexes is a lot more than a googolplex. It might seem that (10^10^10^...^10)^2 is the same as 10^10^10^...^10^2, but in fact a giggol squared is much smaller! (See next entry)

(10^^100)^3

thrargiggol

            This number is still vastly smaller than even E2#100. Simply observe:

(10^^100)^3 = (E1#100)^3 = 10^(3E1#99) < 10^10^(1+E1#98) < ... < E(1+E1#1)#99 

= E11#99 < E100#99 = E2#100

(10^^100)^(10^100)

giggol to the googol

            Surely this is greater than E2#100? Nope, not even close:

(10^^100)^(10^100) = 10^(E1#99 * E100) = 10^10^(100 + E1#98) < 10^10^10^(1+E1#97) 

= E(1+E1#97)#3 < E(1+E1#96)#4 < ... < E(1+E1#1)#99 = E11#99 < E100#99 = E2#100

(10^^100)^(10^10^100)

giggol to the googolplex

            Believe it or not, this is barely an improvement over the last entry. We simply get:

(10^^100)^(10^10^100) = 10^10^(E100 + E1#98) 

< 10^10^10^(1+E1#97) < ... < E11#99 < E100#99 < E2#100

E11#99

            This number is 10^10^10^ ... ^10^10^11 w/99 10s. It's what you would get if you changed the top most exponent in a giggol with 11. Amazingly, despite the fact that this seems like a very minor improvement, it's better than even raising the giggol to the googolplex.

E2#1#2 = E2#100

            This number is 10^10^10^ ... ^10^10^2 w/100 10s. It looks like giggol^2 but it's actually a lot larger. This serves as a lower-bound of megafuga-hundred. This number is also a very small example of a ternary Hyper-E Expression.

100^^100

megafuga-hundred

            In Alistair Cockburn's number system, megafuga-n = n^^n. So naturally megafuga-hundred = 100^^100. It's obvious that this must be larger than 10^^100 (giggol) but probably not by as much as you might think. In fact this number is less than 10^^101. In fact it's even less than E3#100.

E3#100

            This number is 10^10^10^ ... ^10^10^3 w/100 10s. It looks like giggol^3 but it's actually much much larger. This serves as an upper-bound for megafuga-hundred.

10^^101 = E10#100

giggol-plexed

            It is a common rookie response to take whatever the largest named number they know is, and simply have 1 followed by that many zeroes. By the time we get up to numbers of this size however things are moving much much faster, so it turns out to be not all that competitive a response. The reason for the prevalence of this kind of response is probably because f(n) = 10^n is the fastest growing function most people know of.

            In Hyper-E this number can be represented as E1#101, or E10#100. E10#100 acts as a lower-bound for giggol^giggol (see next entry).

(10^^100)^(10^^100)

fzgiggol

            This was a number I used to illustrate how much larger a grangol is than a giggol. Even if you raised a giggol to it's own power, you still would come up vastly short from a grangol. giggol^giggol is also "ever so slightly" greater than 10^^101. This implies that giggol^giggol ~ 10^giggol. However it must be remembered that this is only an approximation. In truth "ever so slightly" is really a huge unimaginable gulf. You would need to raise 10^^101 to the 10^^99th power to get giggol^giggol, so it is really much much much larger in the ordinary sense. The following calculations confirm that giggol^giggol is indeed bounded by 10^^101 and grangol:

giggol^giggol = (E1#100)^(E1#100) = (10^E1#99)^(E1#100) = 10^(E1#99 * E1#100) [Law of Exponents]

= 10^(E1#100 * E1#99) = (10^E1#100)^(E1#99) = (E1#101)^(E1#99) = (10^^101)^(10^^99) > 10^^101

: 10^^101 < giggol^giggol

giggol^giggol = (E1#100)^(E1#100) = 10^(E1#99 * E1#100) = 10^10^(E1#98 + E1#99)

< 10^10^(2E1#99) < 10^10^(10E1#99) = 10^10^(10 * 10^(E1#98)) 

= 10^10^10^(1+E1#98) = E(1+E1#98)#3

< E(1+E1#97)#4 < E(1+E1#96)#5 < ... < E(1+E1#1)#100 = E(1+10)#100 = E11#100 < E100#100 = grangol

((10^^100)^(10^^100))^(10^^100)

giggol to the giggol raised to the giggol

            As amazing as this number sounds, it isn't even as large as E(1+E1#98)#3, let alone E11#100. In my original proof for giggol^giggol < grangol there was an implicit proof that in fact (giggol^giggol)^giggol < E(1+E1#98)#3. This suggests that grangol is much much larger than a giggol than even my initial proof would suggest!

E(1+E1#98)#3

            This number is a relatively exacting upper-bound on giggol^giggol. You can envision it as:

10^10^10^(1+10^10^10^10^ ... ^10^10^10)

            where there are 98 10s inside the parenthesis. In other words, this number is a power tower of tens 101 terms high, with a +1 occurring at the 4th position heading upwards. In comparison with E11#100 (see next entry) this seems like a good bound, but in truth even if you raised giggol^giggol to the power of a giggol you still would come up vastly short of this number!

E11#100

            This number acts as an upper-bound on giggol^giggol. It proves conclusively that it is less than a grangol. Even this upper-bound is actually a huge overestimate.

(100)[3] = f_3(100)

                    This is f_3(100). This number is approximately E32#100, so it lies between a giggol (E1#100), and grangol (E100#100).

E100#100

grangol

            A "grangol" is a number I myself coined in 2011. The name is derived from combining the words "grand" and "googol", thus a grangol is short for "grand googol". It's an example of a number expressible using my Hyper-E notation. Hyper-E notation is a modern equivalent of a notation I devised as a kid. For simple expressions like above let:

Ea#b = 10^10^10^ ... ^10^a w/b 10s

            A grangol is therefore 10^10^10^...^10^10^100 w/100 10s. One way to think of a grangol is as a continuation of the googol tradition. We begin by creating a "googol series". The first member of this series is the eponymous googol, or 10^100. The 2nd member of the series is a googolplex, or 10^10^100. The 3rd member is a googolduplex or 10^10^10^100, and so on. A grangol is defined as the 100th member of the googol series. A grangol turns out to be "just a little" larger than Jonathan Bower's "giggol". In fact, it can be shown that:

giggolgiggol < grangol

            That is, a giggol raised to a giggol is still smaller than a grangol. A giggol in Hyper-E is equal to E1#100. We can work it out as follows:

giggol^giggol = (E1#100)^(E1#100) = (10^E1#99)^(E1#100) =

10^(E1#99*E1#100) < 10^(E1#100*E1#100) = 10^(E1#100)^2 =

10^(10^E1#99)^2 < 10^(10^E1#99)^10 = 10^10^(10*E1#99) =

10^10^(10*10^E1#98) = 10^10^10^(1+E1#98)

= E(1+E1#98)#3 < E(1+E1#97)#4 < E(1+E1#96)#5 < ... etc. ... < E(1+E1#1)#100 =

E(1+10)#100 = E11#100 < E100#100 = grangol

            Thus we conclude that giggol^giggol < grangol.

E1#102 = 10^^102 = <10,102,2>

            This is the smallest power tower of 10s larger than a grangol. As such it acts as an upper-bound, allowing the grangol to be compared to other numbers. To prove grangol < 10^^102 observe:

grangol = E100#100 < E10,000,000,000#100 = E(10^10)#100 = E1#102 = 10^^102.

This number is more imporant than it initially looks, as this forms a basis for upper-bounding Hyper-E Numbers.

10^E100#100 = E100#101

grangolplex

            This was a number I coined to illustrate the inadequacy of the -plex prefix to capture higher level recursions. It is true that Bower's has used the -plex prefix to refer to any type of recursion, however based on the googolplex many have concluded that n-plex always means 10^n. If that is so than a grangolplex is an inadequate name for E100#100#2 (see grangoldex). Following the above definition it follows that grangol-plex = 10^grangol = E(E100#100) = E100#101. 

            This number is greater than 10^^102, but less than grangol^grangol. First observe:

E100#101 > E10#101 = E1#102 = 10^^102

            Since grangolplex = 10^grangol it follows that it is less than grangol^grangol.

(E100#100)^(E100#100)

fzgrangol

            As usual, this number isn't "much larger" than a grangol, at least in terms of power tower height. It must be greater than 10^^102 since grangol^grangol > 10^grangol = E100#101 > E10#101 = E1#102 = 10^^102. However it must be less than 10^^103. This is a little more involved, but can easily be established as follows:

(E100#100)^(E100#100) = 10^(E100#99 * E100#100) = 10^10^(E100#98 + E100#99) 

< 10^10^10^(1+E100#98) 

= E(1+E100#98)#3 < E(1+E100#97)#4 < E(1+E100#96)#5 < ... < E(1+E100#1)#100

= E(1+E100)#100 < E(E101)#100 = E101#101 < E(10^10)#101 = E1#103 = 10^^103

10^^103

            An upper-bound for grangol^grangol.

10^^257

            This is a lower-bound that is commonly used for a Mega since it is the largest integral power tower of 10s which is less than a Mega. This lower-bound is still much much bigger than a grangol, proving that a Mega is also larger.

            It can be written in Hyper-E as E1#257 or E10#256.

E619#256

            This is a more accurate lower bound on the Mega which attempts to narrow down the top most exponent. Written in full it looks like:

10^10^10^10^ ... ... ^10^10^10^10^619 w/256 10s

2[5]

Mega / two in a circle

            This number goes by various names, "Mega", "Zelda", "two in a circle" or "two in a pentagon". It is among the "classic" large numbers along with a googolplex, and Graham's Number. It was first defined by Hugo Steinhaus using his own custom operator notation...(READ MORE). This number is notable for being the largest number on Munafo's Large Number list for a long time. Currently it is still the 3rd largest entry on the list.

E620#256

            This is a more accurate upper-bound on the Mega. It looks like:

10^10^10^10^ ... ... ^10^10^10^10^620 w/256 10s

10^^258

            This is a common upper-bound on the Mega. It is the smallest expression in the form of 10^^N , where N is a positive integer, that is greater than the Mega.

            It can be written in Hyper-E as E1#258 or E10#257 or E10,000,000,000#256.

500^^500

Alternative Interpretation of Blatm's D^^D

                    In the infamous "My Number is Bigger" thread, Gmalivuk pointed out that D^^D was vague and could be interpreted as 13^^13 using hexadecimal or 500^^500 using roman numerals. This latter interpretation is much much larger, though still smaller than Twasbrillig's Power Tower, the 12th competitive entry.

10^^512

Twasbrillig's Power Tower

                    This is the 12th valid entry in the "My Number is Bigger" competition. The original posting of this number was deleted unfortunately but traces of it remain in the responses. This was the first entry to officially break the Tetrational Epoch barrier. This is also the last in an uninterrupted string of larger and larger entries. After this some smaller entries are entered in the competition before this "Large Number in play" is overcome. (See Crazyjimbo's Factorial power tower for entry 13 ).

256^^512

            A weak upper-bound on the Mega based on the Left Associative Tetrates Lemma. Note that we can use the Knuth Arrow Theorem to bound this from above: 256^^512 < (10^^2)^^512 < 10^^514 < 10^^1000. Also 256=2^8 < 2^16 = 2^2^2^2 = 2^^4. Therefore: 256^^512 < (2^^4)^^512 < 2^^516 (Knuth Arrow Theorem). So it is much less than 2^^1000.

2^^1000

This is the second largest entry on Robert Munafo's Numbers List. According to Harvey Friedman this is a lower bound on the number of symbols that would be required to prove the finiteness of TREE(3) using finite arithmetic. Suffice it to say no man nor machine is writing out this proof in this universe. Thankfully ... we are not restricted to proving things in finite arithmetic, because by Kruskal's Tree Theorem we know that TREE(n) is finite

10^^1000

chilialogue / giggolchime

Chilialogue is a number that you must take the common logarithm of 1000 times to reduce it to 1. It may also be called the giggolchime, using the chime modifier on giggol. Chilialogue / giggolchime can be written as E1#1000 in Hyper-E Notation. Expressed as a power tower, this number is about twice as tall as the previous entry.

E3#1#2

E3#1#2 is a simple Ternery Hyper-E expression which does not have a standard name. Solving it we have E3#1#2 = E3#(E3#1#1) = E3#1000. This makes it a little larger than chilialogue. Note however that this value arises naturally from Hyper-E using single digit arguments. This number can be visualized as 10^10^10^10^10^ ... ... ^10^10^10^10^10^3 w/1000 10s. It can also be written in Hyper-E as E1000#999. By the Hyper-E Theorem this makes it a little less than grangolchime.

E1000#1000

grangolchime

Grangol with the chime modifier. This number is larger than giggolchime, as well as the expression E3#1#2. It is smaller than giggoltoll:

E1000#1000 = E(10^3)#1000 = E3#1001 < E10#1001 = E1#1002 << E1#10,000

10^^10,000

myrialogue / giggoltoll

myrialogue is a number so large you need to apply the common logarithm 10,000 times to reduce it to 1. It may also be called the giggoltoll following another naming convention. This entry is a power tower approximately 10 times taller than the last entry, yet we are only getting started with the tetronomical range, let alone numbers described by pentation.

E4#1#2

The next simplest Ternery Hyper-E expression. E4#1#2 = E4#(E4#1#1) = E4#(E4) = E4#10,000. Thus its bigger than E1#10,000 (giggoltoll) and smaller than E10,000#10,000 (grangoltoll).

E10,000#10,000

grangoltoll

The grangol modified by toll. The toll modifier takes all instances of 100 and replaces them with 10,000. Despite how large the first element is becoming it still makes no difference to the second element. 

We can easily show E10,000#10,000 = E(10^4)#10,000 < E(10^10)#10,000 = E1#10,002 << E1#100,000.

10^^100,000

giggolgong

            This number is the result of combining one of Bower's numbers with my -gong suffix. If a googolism, call it g, can be defined as f(100), then g-gong is defined as f(100,000). A giggol = 10^^100, and therefore the giggolgong is 10^^100,000. A giggolgong is a power tower of 10s 100,000 terms high! This number is way way bigger than a Mega, yet it's vanishingly small compare to 256^^(2^256), a naive upper-bound on the Mega.

            In Hyper-E this number can be written as E1#100,000.

E5#1#2

E5#1#2 = E5#(E5#1#1) = E5#100,000 > E1#100,000.

10^^100,001

            A lower bound on a grangolgong. This can be shown to fall between E5#1#2 and E100,000#100,000. First E5#1#2 = E5#100,000 < E10#100,000 = E1#100,001. On the other hand E1#100,001 = E10#100,000 < E100,000#100,000.

E100,000#100,000

grangolgong

            The grangolgong is equal to 10^10^10^ ... ^10^10^100,000 w/100,000 10s. It lies between E10#100,000 = E1#100,001, and E10,000,000,000#100,000 = E1#100,002.

10^^100,002

            A upper bound on a grangolgong. We can show this is greater than a grangolgong. E100,000#100,000 = E(10^5)#100,000 < E(10^10)#100,000 = E1#100,002 = 10^^100,002.

1,000,000^^1,000,000

megafugamillion

                    This is a number that came up in the xkcd forum "My number is, in fact, bigger!", an unofficial sequel to the "My Number is Bigger!" thread. The new competition was initiated by Vytron. This number was defined by "Earthling on Mars" as a part of a larger naive attempt to beat the number <10,10,googol> using iterated power towers.

                    This number is larger than 10^^1,000,000 and so is also larger than 10^^100,002. On the other hand, 1,000,000^^1,000,000 < (10^^2)^^1,000,000 < 10^^1,000,002. At the same time we have 10^^1,000,002 is less than (3^^3)^^1,000,002 < 3^^1,000,005 and therefore much less than 3^^7,625,597,484,987 = 3^^^3.

3^3^3^ ... ^3^3^3 w/7,625,597,484,987 3s

3^^^3

tritri

Jonathan Bowers' tritri is a relatively small pentational number. Expanding it reveals it's formidable size. We have...

3^^^3 = 3^^3^^3 = 3^^3^3^3 = 3^^3^27 = 3^^7,625,597,484,987

It's a power tower of 3's 7,625,597,484,987 terms high. This makes it unfathomably larger than 10^^100,002, but vanishingly small compare to 2^^(2^256). 

2^^(2^256)

            This is a naive upper-bound on the Mega. It is roughly equal to a power tower of 10s E77 terms high. This makes it less than a googol-stack, but more than a grangolgong.

10^^(10100)

googol-stack

            This is another number I found on Cantor's attic. n-stack is defined as 10^^n. In other words, n-stack is a power tower of 10s "n" terms high. Having a power tower of a googol tens certainly seems pretty impressive. Yet this is still small for a pentational number. It is by necessity much larger than a "Mega" since a mega must be less than a power tower of tens only 258 terms high. Yet this number must also be vanishingly small compared to a grangoldex, because a grangoldex is greater than a "stack" of tens a grangol terms high, where a grangol is the 100th member of the googol series!

            Therefore between the googol-stack and the grangoldex must be a vast sea of numbers!

E100#1#2

googoldex

            The googoldex is a number I coined to illustrate just how many kinds of numbers can be named between my numbers using Hyper-E Notation. Hyper-E Notation has the advantage of more easily defining numbers between numbers in other systems. 

            The -dex prefix simply takes some number of the form Ea#b and returns Ea#b#2. So we let googol = E100#1, and so googoldex becomes E100#1#2. What does this mean? Working it out we obtain:

E100#1#2 = E100#(E100#1#1) = E100#(E100) = E100#googol =

EEEEEEEEEEEEE ... ... EEEEEEEEEEEEEEEE100 w/googol Es

= 10^10^10^10^ ... ... ^10^10^10^10^100 w/googol 10s

            In other words, a googoldex is a power tower of 10s a googol terms high topped off with a 100. It's the googolth member of the googol series. 

            It's just "a little" larger than a googol-stack. In fact it's greater than 10^^(googol+1) but less than 10^^(googol+2).

(10100)^^(10100)

megafugagoogol

            A megafugagoogol is only slightly larger than a googoldex. To see why consider the folowing:

googol^^googol > E(10^100)#(10^100-1) = E100#(10^100) = googoldex.

10^((10100)^^(10100))

megafugagoogol-plexed

            This is a non-standard way to interpret "megafugagoogolplex". One can image the "megafuga" being applied before the "plex". To distinguish the cases I coin this number as megafugagoogol-plexed. This number is only slightly larger than a googoldex and vastly smaller than a googolplexidex. This can be demonstrated easily using the Left Associate Polyates Lemma (LAPL):

10^((10^100)^^(10^100)) < 10^((10^10^10)^^(10^100))

= 10^((10^^3)^^(10^100)) < 10^10^^(3+10^100) = 10^^(4+10^100)

< 10^^(10^10^100) = E1#(googolplex) < E100#(googolplex) = E100#(E100#2) = E100#2#2 = googolplexidex

:: megafugagoogol-plexed << googolplexidex

10^^(2^1024)

Limit of break_eternity.js

This is the limit of the break_eternity.js number library. It consists of a power tower of tens whose height is represented with a double-precision floating point number, which has a limit of 2^1024 (in fact this is slightly larger than the largest normal number it can actual represent). This is approximately equal to 10^^(1.797x10^308). It should be noted that at this size the precision must represent an integer, so this value is well defined and doesn't require a real valued tetration to make sense. This entry is mostly notable (to me) for being the very largest Entry of Robert Munafo's famous "Numbers List". So at this point and onwards we are in unexplored territory.

E100#2#2

googolplexidex

            This number is a power tower of 10s a googolplex terms high topped off with a 100.

E100#(1+10^10^100)

googolplexidexiplex

            This number serves a a lower-bound on the megafugagoogolplex. It is also an example of a combinatorial googolism. There is a wealth of numbers that can be derived from various googological systems, an only a tiny fraction of them are ever explicitly stated. These tend to aggregate in certain vicinities, because of the vast differences of power of the different word components.

(E100#2#2)^(E100#2#2)

googolplexidex to the power of a googolplexidex

            This number is incredibly close to a megafugagoogolplex, yet it is still slightly smaller.

E(1+10^100)#(10^10^100)

            This number is an upperbound on googolplexidex^^2.

E(100+10^100)#(10^10^100)

            A very accurate lower-bound on a megafugagoogolplex.

(10^10^100)^^(10^10^100)

megafugagoogolplex

            This number is derived from the work of Alistair Cockburn. It combines his megafuga- prefix with the number googolplex. n-plex = 10^n while megafuga-n = n^^n. Note that the definition here is actually ambiguous. Is a megafugagoogolplex equal to megafuga(googolplex) or plex(megafuga(googol). That is, we can read it grammatically as meaning either megafuga-"googolplex" or as "megafugagoogol"-plex. These result in different numbers (See megafugagoogol-plexed). The original intent of Cockburns work however is that the prefixes are being applied after the suffixes. Thus a megafugagoogolplex is intended to mean googolplex^^googolplex. This number has come up independently from a few sources , usually as an example of the largest kind of number the average person would think of to try to trump Graham's Number (psst ... it doesn't even come close. See Graham's Number far below). That being the case it's nice to have a name, any name, for this number.

            This number is pretty insanely huge, although it has more to do with the height of the stack than the terms being a googolplex. It confers just enough benefit so that it goes slightly past a googolplexidex. A googolplexidex is a power tower with a googolplex+1 terms, where as a megafugagoogolplex has only a googolplex terms, but the megafugagoogolplex ends up being ever so slightly larger (from a googologist's perspective) mainly because of the leading exponent. It can be shown that megafugagoogolplex lies between E100#(googolplex+1) and E100#(googolplex+2), and is greater than googolplexidex^googolplexidex. For a worked out proof click here.

E(101+10^100)#(10^10^100)

            This is an accurate upper-bound on the megafugagoogolplex.

E100#(2+10^10^100)

googolplexidexiduplex

            This number serves as a "weak" upper-bound on a megafugagoogolplex. It's also an example of a combinatorial googolism. 

E100#3#2

googolduplexidex

            This number is a power tower of 10s a googolduplex terms high topped off with a 100. This number is still vastly smaller than a giggolplex.

E100#4#2

googoltriplexidex

E100#(googoltriplex) = E100#(10^10^10^10^100). 

E100#5#2

googolquadriplexidex

E100#(googolquadriplex) = E100#(E100#5).

E100#6#2

googolquintiplexidex

E100#(googolquintiplex) = E100#(E100#6).

E100#7#2

googolsextiplexidex

E100#(googolsextiplex) = E100#(E100#7).

E2#9#2

This number looks like it's equal to E100#8#2, but in fact this is not the case. Expanding we have E2#9#2 = E2#(E2#9) = E2#(E100#8) < E100#(E100#8) = E100#8#2. Thus we prove E2#9#2 < E100#8#2 even though E2#9 = E100#8. The number E2#9#2 can be used as an approximation of googolseptiplexidex using the Hyper-E Number Format, in which the arguments must be less than or equal to 10.

E100#8#2

googolseptiplexidex

E100#(googolseptiplex) = E100#(E100#8). E100#(E100#8) = E(10^2)#(E100#8) < E(10^10)#(E100#8) = E1#(2+E100#8) < E1#(E101#8) < E1#(E(10^10)#8) = E1#(E1#10) = E1#(E1#(E1#1)) = E1#1#3 = 10^^^3 = 10^^10^^10 = triataxis. Thus a googolseptiplexidex is smaller than a triataxis, the limit of the Tetronomical Epoch ...

10^^(10^10^10^10^10^10^10^10^10^9)

E1#(E9#9)

This number makes the limit of break_eternity.js seem small by comparison. It's just a little bit smaller than a triataxis.

E9#9#2

... only ... wait a minute ... with Hyper-E Notation we can actually get even closer to a triataxis than even that! Observe: E9#9#2 = E9#(E9#9) < E10#(E9#9) = E1#(1+E9#9) < E1#(E10#9) = E1#(E1#10) = E1#(E1#(E1#1)) = E1#1#3 = 10^^^3 = 10^^10^^10 = triataxis.

10^^10^^10

triataxis

                    This number is 10^^^3. It is also the 13th official competitor in the "My Number is Bigger" competition, and the 15th valid number. This number is the 3rd entry by Gmalivuk, the starter of the competition. It is after this number that Rodan "shuffles out". 

It also happens to be a number I call triataxis. This means you have to apply the "taxarithmic" function (superlogarithm) 3 times to reduce it to 1. The reason I call it the "taxarithmic" function is because I think of it as classifying numbers into taxonomic categories (similar to Robert Munafo's Number classes concept). Let taxa(x) be the taxarithmic function defined by taxa(x) = taxa(log(x))+1 : x > 1 , taxa(x) = 0 : x <= 1. It follows from the definition that taxa(10^^n) = n. Thus we have that: taxa(taxa(taxa(triataxis))) = taxa(taxa(taxa(10^^10^^10))) = taxa(taxa(10^^10)) = taxa(10) = taxa(1)+1 = 0+1 = 1.

E100#9#2

googoloctiplexidex

E100#10#2

googolnoniplexidex

E100#11#2

googoldeciplexidex

10^^10^^100

giggolplex

            The giggolplex is a number coined by Jonathan Bower's as an extension of the googol naming conventions. A giggolplex is a power tower of 10s a giggol terms high, where a giggol is itself a power tower of 10s 100 terms high. It can be notated as:

giggolplex = 10^10^10^ ... ^10^10^10 w/giggol 10s

E100#100#2

grangoldex

            Here's a HUGE number, very similar in spirit to the once great googolplex. A grangoldex is a power tower of 10s a grangol terms high, with a 100 on top of all this! Another way to think about it is that a grangoldex is the grangolth member of the googol series. We can also notate it as:

grangoldex = 10^10^10^ ... ^10^10^10^100 w/grangol 10s

            This number is just slighly larger than a giggolplex. This can be seen since giggol < grangol, it follows that a grangoldex has more 10s than a giggolplex.

E1#102#2 = 10^^10^^102

An elementary upperbound on the grangoldex. Proving this is fairly straightforward:

E100#100#2 = E100#(E100#100) < E(10^10)#(E100#100) = E1#(2+E100#100)

To continue we simply observe that a+Eb@ < E(b+1)@ provided a is smaller than or equal to Eb@, regardless of any other factors, because every expression of the form Eb@ must be a power of 10, and if a <= Eb@ we have a+Eb@ <= 2Eb@ < 10Eb@ <= E(b+1)@. The upshot of this is we can always get an upperbound just by increasing the top exponent by 1. So we have:

E1#(2+E100#100) < E1#(E101#100) < E1#(E(10^10)#100) = E1#(E1#102) = E1#102#2

Thus E100#100#2 < E1#102#2. It shouldn't be too hard to see that this process can be iterated an arbitary number of times, carrying changes upwards, implying that:

E100#100#n < E1#102#n

This will be important for later.

(1,000,000^^1,000,000)^^(1,000,000^^1,000,000)

mungo

                    This is a googolism coined by "Earthling on Mars" as part of a naive attempt to beat <10,10,googol> using power towers. He describes 1,000,000^^1,000,000 as a "mega" and mungo as mega^^mega. It turns out however not be any faster an iteration than pentation. From the Knuth-Arrow theorem (see my paper "A Theorem for Knuth-Arrows") it follows that...

(1,000,000^^1,000,000)^^(1,000,000^^1,000,000) < 1,000,000^^(1,000,000+1,000,000^^1,000,000)

< (10^^2)^^(1,000,000+1,000,000^^1,000,000) < 10^^(1,000,002+1,000,000^^1,000,000)

< 10^^(1,000,002+10^^1,000,002) << 10^^10^^1,000,003

This means this number is still massively smaller than 10^^10^^10^^100 or giggolduplex. On the other hand, grangoldex = E100#100#2 = E100#(E100#100) < E1#(2+E1#102) < E1#(E1#103) = 10^^10^^103, and this is much smaller than (1,000,000^^1,000,000)^^(1,000,000^^1,000,000), so we know a mungo is larger than grangoldex. Thus we can say...

grangoldex << mungo << giggolduplex

E100#1#3 = E100#(10^100)#2

googoldudex

3[5]

triton

    Triton is the name I coined for "3 in a circle" from Steinhaus's circle function. This number can be approximated fairly closely in ExE as E40#2#3. This means it falls between googoldudex and googolplexidudex. This is a far far far larger number than mega, or "2 in a circle". While Steinhaus named 2[5] mega and 10[5] megiston he did not provide any other names for numbers. Moser later defined a much much bigger number now simply called Moser.

E100#2#3 = E100#(10^10^100)#2

googolplexidudex

E100#3#3 = E100#(E100#3)#2

googolduplexidudex

E100#4#3 = E100#(E100#4)#2

googoltriplexidudex

E100#5#3 = E100#(E100#5)#2

googolquadriplexidudex

E100#6#3 = E100#(E100#6)#2

googolquintiplexidudex

E100#7#3 = E100#(E100#7)#2

googolsextiplexidudex

E100#8#3 = E100#(E100#8)#2

googolseptiplexidudex

E100#9#3 = E100#(E100#9)#2

googoloctiplexidudex

10^^10^^10^^10

tetrataxis

The tetrataxis is ten pentated to the fourth, 10^^^4. It is therefore one of the basic Knuth arrow outputs. It can be proven with very little difficultly that E100#10#3, which can be named googolnoniplexidudex, is actually "slightly" larger, though it's difficult to say exactly how (numbers at this scale already have no easy way to compare them in ordinary terms). First we can write tetrataxis in Hyper-E Notation as E1#1#4. From there we can show that this is equal to E1#10#3. To do so, we simply note that if @ = 'E1#' then E1#1#4 may be expanded to @@@(@1). @1 = E1#1 = E1 = 10. So we have @@@10. This is E1#(E1#(E1#10)) which can be written compactly as E1#10#3. From here we can use the Hyper-E lemma that if we increase any argument in a Hyper-E expression, the resulting expression is larger. Thus we have E1#10#3 < E100#10#3. Thus we conclude: tetrataxis = 10^^^4 = E1#10#3 < E100#10#3 = googolnoniplexidudex. Therefore tetrataxis is smaller than googolnoniplexidudex. With a little bit more effort it can be shown that E1#10#3 > E100#9#3, the previous entry.

E100#10#3 = E100#(E100#10)#2

googolnoniplexidudex

E100#11#3 = E100#(E100#11)#2

googoldeciplexidudex

10^^10^^10^^100

giggolduplex

A Bowerism found in the "Giggol Group".

E100#100#3

grangoldudex

            grangoldudex = E100#100#3 = E100#(E100#100#2) 

= E100#grangoldex = EEE...EEE100 w/grangoldex Es.

            The grangoldudex is smaller than 10^^^5. This can be seen as follows:

E100#100#3 = E100#(E100#(E100#100)) < E100#(E100#(E10,000,000,000#100)) = E100#(E100#(E1#102)) 

< E100#(E1#(2+E1#102)) < E1#(2+E1#(2+E1#102)) < E1#(E1#(3+E1#102)) < E1#(E1#(E1#103)) = E1#103#3 

< E1#(10^^10)#3 = E1#1#5 = 10^^^5

E1#102#3

An upperbound on grangoldudex, and part of a special upperbounding sequence. Again we can always move up while "rounding things up" to get upperbounds. So for grangoldudex we have:

E100#(E100#(E100#100)) < E1#(E101#(E100#100)) < E1#(E1#(E101#100)) 

< E1#(E1#(E1#102)) = E1#102#3.

This pattern continues.

((1,000,000^^^2)^^^2)^^^2

humungo

                    This googolism was coined by "Earthling on Mars"  as part of the "My number is, in fact, bigger" thread.

Bounding it from below we have ...

humungo > ((10^^1,000,000)^^^2)^^^2 

> (10^^10^^1,000,000)^^^2 > 10^^10^^10^^1,000,000 = E1#1,000,000#2

Bounding it from above we have ...

humungo < 1,000,000^^^6 < (3^^^2)^^^6 < 3^^^8

Since humungo is less than 3^^^8 we know it's less than 3^^^10.

E100#1#4 = E100#(10^100)#3

googoltridex

E100#2#4 = E100#(E100#2)#3

googolplexitridex

E100#3#4 = E100#(E100#3)#3

googolduplexitridex

5^^^5

~ E2184#3#4

Tripent Jr.

           Bowers' original value for a tripent, which is "five pentated to the fifth". In the old array notation this was written as {5,5,5}, hence "tri"(3)+"pent"(5), that is, three fives. In the new array notation this is equivalent to <5,5,3>. We can actually bound this quite accurately in Hyper-E notation as about E2184#3#4, the exact value being only "slightly" larger. This allows us to easily see this is above E100#3#4 yet below E100#4#4, the latter being approximately E(10^100)#3#4. Hyper-E thus allows us to pretty precisely measure different notations against each other. 

E100#4#4 = E100#(E100#4)#3

googoltriplexitridex

E100#5#4 = E100#(E100#5)#3

googolquadriplexitridex

E100#6#4 = E100#(E100#6)#3

googolquintiplexitridex

E100#7#4 = E100#(E100#7)#3

googolsextiplexitridex

E100#8#4 = E100#(E100#8)#3

googolseptiplexitridex

E100#9#4 = E100#(E100#9)#3

googoloctiplexitridex

E100#10#4 = E100#(E100#10)#3

googolnoniplexitridex

E100#11#4 = E100#(E100#11)#3

googoldeciplexitridex

E100#100#4

grangoltridex

A grangoltridex = E100#grangoldudex. That is, a power tower of 10s grangoldudex terms high topped off with 100. What would Milton and Kasner say to that!

E1#102#4

An upperbound on a grangoltridex. It is equal to 10^^10^^10^^10^^102. This is trivially smaller than ...

E100#1#5 = E100#(10^100)#4

googolquadridex

E100#2#5 = E100#(10^10^100)#4

googolplexiquadridex

E100#3#5

googolduplexiquadridex

E100#4#5

googoltriplexiquadridex

E100#5#5

googolquadriplexiquadridex

E100#6#5

googolquintiplexiquadridex

E100#7#5

googolsextiplexiquadridex

E100#8#5

googolseptiplexiquadridex

E100#9#5

googoloctiplexiquadridex

E100#10#5

googolnoniplexiquadridex

E100#11#5

googoldeciplexiquadridex

E100#100#5

grangolquadridex

grangolquadridex = E100#grangoltridex.

E1#102#5

An upperbound on a grangolquadridex.

E100#1#6 = E100#(10^100)#5

googolquintidex

E100#2#6 = E100#(10^10^100)#5

googolplexiquintidex

E100#3#6

googolduplexiquintidex

E100#4#6

googoltriplexiquintidex

E100#5#6

googolquadriplexiquintidex

E100#6#6

googolquintiplexiquintidex

E100#7#6

googolsextiplexiquintidex

E100#8#6

googolseptiplexiquintidex

E100#9#6

googoloctiplexiquintidex

E100#10#6

googolnoniplexiquintidex

E100#11#6

googoldeciplexiquintidex

E100#100#6

grangolquintidex

grangolquintidex = E100#grangolquadridex

E1#102#6

An upperbound on a grangolquintidex.

E100#1#7 = E100#(10^100)#6

googolsextidex

E100#2#7 = E100#(10^10^100)#6

googolplexisextidex

E100#3#7

googolduplexisextidex

E100#4#7

googoltriplexisextidex

E100#5#7

googolquadriplexisextidex

E100#6#7

googolquintiplexisextidex

E100#7#7

googolsextiplexisextidex

E100#8#7

googolseptiplexisextidex

E100#9#7

googoloctiplexisextidex

E100#10#7

googolnoniplexisextidex

E100#11#7

googoldeciplexisextidex

E100#100#7

grangolsextidex

grangolsextidex = E100#grangolquintidex = E100#(E100#100#6)

E1#102#7

A tight upperbound on a grangolsextidex.

E100#1#8 = E100#(10^100)#7

googolseptidex

E100#2#8 = E100#(10^10^100)#7

googolplexiseptidex

E100#3#8

googolduplexiseptidex

E100#4#8

googoltriplexiseptidex

E100#5#8

googolquadriplexiseptidex

E100#6#8

googolquintiplexiseptidex

E100#7#8

googolsextiplexiseptidex

E100#8#8

googolseptiplexiseptidex

E100#9#8

googoloctiplexiseptidex

E100#10#8

googolnoniplexiseptidex

E100#11#8

googoldeciplexiseptidex

E100#100#8

grangolseptidex

    grangolseptidex is just below 3 pentated to the 10th. It's actually a pretty good lower bound.

E1#102#8

An upperbound on a grangolseptidex. To show this is less than 3^^^10 we note this is equal to 10^^10^^10^^10^^10^^10^^10^^10^^102. This is smaller than:

27^^27^^27^^27^^27^^27^^27^^27^^102

27 = 3^^2. Using the knuth arrow theorem this means its smaller than:

3^^(2+3^^(2+3^^(2+3^^(2+3^^(2+3^^(2+3^^(2+3^^104)))))))

By ascending each 2 to get a height of 1 greater we get this is all less than:

3^^3^^3^^3^^3^^3^^3^^3^^105

which is 3^1#105#8. 3^^3 = 7,625,597,484,987 >> 105. This means this last Hyper-E expression is less than 3^1#(3^^3)#8 = 3^1#1#10 = 3^^^10. Thus we have proven E1#102#8 is less than ...

3^^^10

Blatm's Pentational Number

                    The 14th competitor in the "My Number is Bigger" competition, and the 16th valid number. This number was entered by Blatm by reversing the order of the arguments of Gmalivuk's number. We are now well into pentational numbers. At this point elementary arithmetic expressions with exponents and factorials can no longer compete.

4^^^10

Xooll Shrug's

                    The 15th competitive entry in the "My Number is Bigger" competition, and the 17th valid entry. Xooll entered in response to Blatm's Pentational Number. Xooll typed *shrug* after it, as if to say, what's the big deal? But this response isn't terribly competitive and the real competition has only begun.

10^^^10 = E1#1#10

dekataxis

            This number is equal to 10^^10^^10^^10^^10^^10^^10^^10^^10^^10. To envision this number imagine Stage 1 as "10", Stage 2 as "10^10^10^10^10^10^10^10^10^10", Stage 3 as "10^10^ ... ^10^10" w/Stage 2 10s, ... and go all the way to Stage 10. This massive number is bigger than even the grangoldudex, but still smaller than the Megiston.

            In Hyper-E this can be written as E1#1#10 which is also equivalent to E1#10#9.

(10^^^10)!

Ended's Salad Factorial

                    Technically this is the 16th competitive entry in the "My Number is Bigger" competition, and the 18th valid entry. It was entered by User Ended. Most of the strength of the number comes from pentation. At this point the numbers are so big that adding a factorial is so negligible that we can ignore it as a "salad factorial". To understand this, realize that at this scale N! ~ 10^N. Furthermore we have 10^^^10 = 10^^(10^^^9). So 10^^^10 is a power tower of 10s 10^^^9 terms high. 10^^^9 is an inconceivably vast number. From this we can gather that (10^^^10)! ~ 10^(10^^^10) = 10^^(1+10^^^9) ~ 10^^(10^^^9) = 10^^^10. So it has virtually no effect.

E10#9#9

This is a Hyper-E expression that is approximately (but not exactly) equivalent to E1#1#10. It's listed here to illustrate some principles of comparing Hyper-E Numbers. We can prove that this number is in fact larger than E1#1#10. To do so we first write E1#1#10 as E1#10#9. Next we expand both using the alternative recursive law:

E1#10#9 = E1#(E1#10)#8

E10#9#9 = E10#(E10#9)#8

We note here that E1#10 = E10#9 so this means the last two arguments of these numbers are identical, but the first argument is larger for E10#9#9 than E1#10#9, and so E10#9#9 is surprisingly larger. How much larger? In googological terms its extremely close and barely distinguishable. When expanded we have:

E1#(E1#10)#8 = E1#(E1#(E1#(E1#(E1#(E1#(E1#(E1#(E1#10))))))))

yet ...

E10#(E1#10)#8 = E10#(E10#(E10#(E10#(E10#(E10#(E10#(E10#(E1#10))))))))

Each of these 10s only has the effect of adding 1 to the next power towers height as opposed to using 1s. For example E10#(E1#10) = E1#(1+E1#10), which with a power tower this tall barely makes a difference. This then becomes the height of the next power tower which also gets topped off with one additional 10, and so on. The upshot is that these two numbers are incredibly close yet E10#9#9 is just a little bit larger. This still ends up becoming much more than a factorial. Why? Because that only increases the height of the last power tower by 1. In this case it would be equivalent to adding 1 at the first power tower, which will have a cascading effect, leading to a much larger end result. And yet even this is really nothing next to ...

10^^^11

            This number serves as a weak lower-bound for the Megiston.

10[5]

Megiston / ten in a circle

            The Megiston is the lesser known of the two numbers Hugo Steinhaus defined with his circle notation. It is much much larger than a Mega, taking advantage of the full power of the circle operator, which is roughly on par with pentation. This number is much more difficult to bound than the Mega, due to various technical difficulties. It isn't too difficult however to show that it must lie somewhere between 10^^^11 and 10^^^12.

            *Assuming side note: Bowers' incorrectly calls this number megaston on his infinity scrapers page.

10^^^12

This number serves as a weak upper-bound for the megiston.

ten pentated to twelve

10^^^100 

gaggol

                This is Jonathan Bowers' gaggol, defined as 10^^^100. This makes it a very large pentational number and a very small hexational number. This number is larger than a megiston, but is "slightly" smaller than a greagol. In Hyper-E it can be expressed as E1#1#100.

100^^^100

                This number serves as a benchmark for the largest pentational number. Although this designation is arbitrary, a pentational number is usually understood as anything of the form a^^^b where a and b are relatively small arguments and the result is a number not already included in a smaller class of numbers. Hence if we let 100 be the limit of a "relatively small" argument, then 100^^^100 is the largest pentational number. Interestingly, this number is not that much larger than a gaggol or 10^^^100, relatively speaking. Next up ... the Ackermann Numbers ...

E3#100#99

                This number serves as an upperbound on 100^^^100. Consider that 100^^n < E3#n. Therefore 100^^100 < E3#100, 100^^^3 = 100^^100^^100 < 100^^E3#100 < E3#(E3#100) = E3#100#2, 100^^^4 = 100^^100^^^3 < 100^^E3#100#2 < E3#(E3#100#2) = E3#100#3 ... and in general 100^^^n < E3#100#(n-1). Since E3#100#99 < E100#100#100 it follows that 100^^^100 < greagol. It can also be shown that 100^^^100 < E3#100#99 < 10^^^101. For a more detailed proof click here.

10^^^101

                This serves as both an upper-bound on 100^^^100 and a lower bound on a greagol. The lower-bound is easier to demonstrate. Simply observe that 10^^^101 = E1#1#101 = E1#(E1#1#100) = E1#(E1#(E1#1#99)) = E1#(E1#1#99)#2 = E1#(E1#(E1#(E1#1#98))) = E1#(E1#1#98)#3 = ... = E1#(E1#1#1)#100 = E1#10#100 < E100#100#100 = greagol.

E100#100#100

greagol

                A greagol, short for "great googol", is the 100th member of the grangol series. It is larger than and comparable to Jonathan Bowers' gaggol.

E1#102#100

                This number is larger than a greagol, but less than 10^^^102. Hyper-E allows one to express more intermediate values than Knuth Up-arrow notation. 

10^^^102 = E1#1#102

 This is an upperbound on a greagol. In Hyper-E it can be written E1#1#102. This is one of the intervening steps in the proof that greagol << Folkman's Number.

Note that we can show that E1#102#100 < E1#1#102 easily as follows:

E1#102#100 < E1#(10^^10)#100 = E1#1#102.

16^^^102

                This is another step in the proof greagol << Folkman's Number.

2^^^408

                This is the final step in the proof greagol << Folkman's Number. By converting the base of the pentation to 2, it's made immediately apparent that this number must be less than Folkman's Number of 2^^^(2^901). Simply consider 2^^^408 < 2^^^512 = 2^^^(2^9) << 2^^^(2^901)

2^^^(2^901)

Folkman's Number

            This moderately sized Ackermann class number was mentioned in the same article by Martin Gardner where he introduced the world to "Graham's Number" (See article here). Folkman was looking for a graph containing no K4s that forces a monochromatic K3 when it's two-colored. He devised an example of such a graph ... but it would contain 2^^^(2^901) points! This number is insanely large. Yet it's still smaller than G(1) of Graham's Number. Folkman's Number is somewhere between a greagol and G(1). Roughly speaking, the reason is because a greagol ~ 2^^^100 (actually larger) where as G(1) ~ 2^^^(3^^7,625,597,484,987) (actually larger). For a full proof click here.

G(1)

3^^^^3

                This is 3 hexated to the 3rd. Evaluating  it we have:

3^^^^3 = 3^^^3^^^3 = 3^^^3^^3^^3 = 3^^^3^^3^3^3 =

3^^^3^^3^27 = 3^^^3^^7,625,597,484,987 =

3^^^3^3^3^3^ ... ^3^3^3^3 w/7,625,597,484,987 3s after 3^^^ =

3^^3^^3^^3^^3^^ ... ^^3^^3^^3^^3^^3

w/3^3^3^ ... ^3^3^3 3s

w/7,625,597,484,987 3s

            To imagine this, let stage 1 = 3. Let stage 2 = 3^3^3 or 7,625,597,484,987, let stage 3 = 3^3^ ... ^3^3 w/7,625,597,484,987 3s, and in general each new stage is a power tower of 3s with the previous stage number of terms. 3 hexated to the 3rd is Stage 3^3^3^ ...^3^3^3 w/7,625,597,484,987 3s. This number is also G(1), the first member of graham's sequence (See G(64)).

((...((1,000,000^^^2)^^^2)...)^^^2)^^^2

w/((1,000,000^^^2)^^^2)^^^2-1 "^^^2"s

Earthling on Mars Number

                This is the final form of Earthling on Mars's Number. This was his attempt to come up with a number larger than <10,10,googol> using power towers. This number however can be demonstrated to be in the hexational range, much much smaller than <10,10,googol>. Firstly we can observe that...

Earthling on Mars Number < 1,000,000^^^(2*1,000,000^^^6) < 10^^^(2+2*1,000,000^^^6) 

< 10^^^(3*1,000,000^^^6) < 10^^^(3*10^^^8) < 10^^^(10*10^^^8) < 10^^^(10^^10^^^8)

= 10^^^10^^^9 < 10^^^10^^^10 = 10^^^^3 < 10^^^^10 = <10,10,4>

                    So Earthling on Mars Number is less than even <10,10,4>. In fact it's less than 10^^^10^^^10 making it smaller than a gaggolplex. On the other hand we have...

Earthling on Mars Number > 10^^^10^^10^^10^^10 = 10^^^10^^^4 > 3^^^3^^^3 = 3^^^^3.

So the Earthling on Mars Number is bigger than 3^^^^3 or G(1).

10^^^10^^^100

gaggolplex

This number is massively larger than G(1), yet at this stage it starts becoming more obscure why. The reason is because 3^^^3 (tritri) is vastly smaller than 10^^^100 (gaggol). Therefore 3^^^(3^^^3) << 10^^^10^^^100.

E100#100#100#2

greagolthrex

This is the first number to introduce the -threx operator. "three" + "plex" = "threx". For a number of the form n = Ea#b#c, threx-n = Ea#b#(Ea#b#c). 

Therefore a greagolthrex = E100#greagol = E100#(E100#100#100) = E100#100#100#2.

E1#1#102#2

An important upperbound on a greagolthrex. Recall that E100#100#n < E1#102#n. From here we prove E100#100#100#2 < E1#1#102#2.

E100#100#100#2 = E100#(E100#100#100) < E1#(2+E100#100#100) 

< E1#(E101#100#100) < E1#(E1#102#100) < E1#(E1#1#102) = E1#1#102#2. 

10^^^10^^^10^^^100

gaggolduplex

Another Bowerism on the Infinity scrapers' page. In E# this can be written as E1#1#100#3 making it definitely smaller than E100#100#100#3 (though googologically in the same neighborhood basically). It is still vastly larger than a greagolthrex. Just observe:

E100#100#100#2 = E100#100#greagol < E100#(E100#1#1)#greagol = E100#1#(1+greagol)

E100#(E100#1#greagol) < E(10^10)#(E100#1#greagol) = E1#(2+E100#1#greagol)

= E1#(2+E100#(E100#(E100#(E100#(E100#(... E100#(E100#1)...))))))))

E1#(E1#(E1#(E1#(E1#(E1#(E1#(E1#(E1#( ... E1#(E101#1) ... ))))))))))

E101#1 < E(10^10)#1 = E1#3

so we get an upperbound of:

E1#3#(1+greagol)

which is less than E1#(E1#1#1)#(1+greagol) = E1#1#(2+greagol) = E1#1#(2+E100#100#100)

we can again ascend the 2 up through the various power towers eventually reaching the innermost level...

E101#100 which is still less than E(10^10)#100 = E1#102 so we get...

E1#1#(E1#102#100) and since 102 < 10^^10 = 10^^^2 = E1#1#2 we have:

E1#1#(E1#(E1#1#2)#100) = E1#1#(E1#1#102)

which is E1#1#102#2 which is less than E1#1#100#3 since this is equal to E1#1#gaggol#2.

E100#100#100#3

greagolduthrex

           greagol threxed twice.

E1#102#100#3

         An upperbound on a greagolduthrex.

E1#1#102#3

10^^^10^^^10^^^102

An even larger upperbound on a greagolduthrex.

E1#1#1#5

10^^^^5 = <10,5,4>

An even weaker upperbound on a greagolduthrex. This one is a simple upperbound in up-arrow notation, which brings us to ...

 <10,100,4>

E1#1#1#100

geegol

     A Bowerism based an extension of the googol. This is the forth in the sequence: googol, giggol, gaggol, geegol. Unlike some of Bowers' other numbers, this one did not shift between Bowers' older array notation and newer array notation. Bowers simply decreased the third argument accordingly. On his original site it was denoted as {10,100,6} (10 hexated to the 100th). On the new site it's <10,100,4>. I'll be using the angle brackets to denote Bowers new array notation, and curly braces to denote the old notation. In Hyper-E geegol can be expressed exactly as E1#1#1#100. This makes it "slightly" smaller than ...

10^^^^101 = E1#1#1#101 = E1#1#10#100

         This one might require a little explanation. Note that using the alternative expansion:

E1#1#1#101 = E1#1#(E1#1#1)#100 = E1#1#(E1)E100 = E1#1#10#100. This is analogous to the way E1#101 is larger than a giggol and smaller than a grangol, and E1#1#101 is larger than a gaggol yet smaller than a greagol. The proofs in these cases are the same: E1#101 = E10#100 < E100#100 , E1#1#101 = E1#10#100 < E100#100#100.

E100#100#100#100

gigangol

            A gigangol, short for "gigantic googol", is the 100th member of the greagol series.

10^^^^102 = E1#1#1#102

         A modest upperbound on a gigangol. This follows from E100#100#100#n < E1#1#102#n. So E100#100#100#100 < E1#1#102#100 < E1#1#(10^^^10)#100 = E1#1#1#102.

<10,<10,100,4>,4>

E1#1#1#100#2

geegolplex

           The plex version of the geegol. It's worth noting that after the gaggolduplex, Bowers' never has more than one plex version for any number. At this point he races through the next couple of generations of googol with two items each.

E100#100#100#100#2

gigangoltetrex

           The "tetrex" version of a gigangol. tetrex is the gigangol, as threx is to greagol, and dex is to grangol. These special operators allow the next level of recursion.

E100#100#100#100#3

gigangoldutetrex

           I like to have at least two additional version after introducing a new number name. This brings us to the next level ...

{7,7,7} = <7,7,5>

Trisept Jr.

         The original "trisept" or {7,7,7}. It's now written as <7,7,5> in Bowers' new array notation. This value can be approximately fairly accurately in Hyper-E as E695,974#6#6#6#6. This places it above gigangoldutetrex but still below a gygol.

<10,100,5>

E1#1#1#1#100

gygol

           Originally called a gygol, this is the fifth level of the sequence: googol, giggol, gaggol, geegol, gygol. In the originally array notation this was written as {10,100,7}. In the new notation it equals exactly <10,100,5>. At this point with 5 up-arrows this is inconvenient to write in up-arrow notation, so Bowers array notation becomes more useful.

E100#100#100#100#100

gorgegol

            The gorgegol, short for "the gorged googol", is the 100th member of the gigangol sequence.

<10,<10,100,5>,5>

E1#1#1#1#100#2

gygolplex

           The plex version of a gygol. Interestingly, although Bowers frequently uses this type of recursion, it doesn't come very naturally to array notation. He typically uses a name in place of another array expression describing gygolplex as {10,gygol,7} or <10,gygol,5>. However I prefer to avoid the potential ambiguity given that some names have different meanings depending on which version of array notation we are talking about. In any case Hyper-E allows us to easily apply additional levels of recursion by simply appending another argument at the end. 

E100#100#100#100#100#2

gorgegolpentex

           gorgegol+pentex.

E100#100#100#100#100#3

gorgegoldupentex

           gorgegol+pentex+pentex.

<10,100,6>

goggol

         Bowers sixth googol variant. a goggol is comparable to a gulgol ...

E100#100#100#100#100#100

gulgol

            The gulgol, short for the *gulp* googol, is the 100th member of the gorgegol series.

<10,<10,100,6>,6>

goggolplex

           The obligatory "plex" version of a goggol.

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

gulgolhex

           hexa + plex = "hex". gugol may be appended with an arbitrary number of "hexes". 

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

gulgolduhex

           gugol + hex + hex = gulgol(2)hex = gulgolduhex. On to the next group/regiment ...

<10,100,7>

gagol

           Not to be confused with a "gaggol". The seventh and last of the official Bowers googol variants. A gagol is comparable to a gaspgol ...

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

E100##7

gaspgol

            The gaspgol, short for "gasp googol", is the 100th member of the gulgol series.

<10,<10,100,7>,7>

gagolplex

         The plex version of a gagol. This concludes Bowers most basic googolisms ...

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

gaspgolheptex

           "hepta" + "plex" = "heptex". We can append as many heptexes to gaspgol as needed/wanted.

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

gaspgolduheptex

           gaspgol + heptex + heptex = gaspgol(2)heptex = gaspgolduheptex.

E1#1#1#1#1#1#1#5

<10,5,8>

         An upperbound on a gaspgolduheptex. Using a "zipper"-trick we can get rid of all the 100s and show it's less than this value ... which leads us to ...

{10,10,10} = <10,10,8>

E1#1#1#1#1#1#1#10

tridecal Jr.

           Bowers' original value for the "tridecal". In the original array notation this would be {10,10,10}, or three-tens, hence the name "tri"(3) + "decal"(10). However when Bowers changed his notation this old value shifted to <10,10,8>. The new value is <10,10,10>. Tridecal Jr. is just a little smaller than a ginorgol, and would also be smaller than a hypothetical 8th Bowers' googol variant ...

<10,100,8>

E1#1#1#1#1#1#1#100

         This would be the value for a hypothetical 8th Bowers' googol variant. It would be comparable and slightly smaller than a ginorgol ...

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

ginorgol

            The ginorgol, short for "ginormous googol", is the 100th member of the gaspgol series.

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

ginorgoloctex

         "octa" + "plex" = "octex". We may append octexes to ginorgols, forming the ginorgol sequence ...

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

ginorgolduoctex

ginorgol-octex-octex.

<10,100,9>

         A hypothetical 9th Bowers' googol variant to match with 

E100##9

gargantuul

           The nineth of my elementary googol extensions. It's the 100th member of the ginorgol sequence. A gargantuul comes from "gargantuan" + "googol". Approximately equal to 10<9>102. We are getting close to Bowers' new tridecal ...

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

gargantuulennex

           "enna" + "plex" = "ennex". Repeated applications of ennex to gargantuul produces the gargantuul sequence. Roughly equal to 10<9>10<9>102 which is less than 10<9>10<9>10<9>10 = 10<10>4 = <10,4,10>. Thus this number is just shy of a tridecal ...

<10,10,10>

10^^^^^^^^^^10

Tridecal

                    This number can also be written as <10,10,10> using linear array notation. This googolism was coined by Jonathan Bowers. This Number is also Bowers' benchmark for an "Infinity Scraper"

(10^^^^^^^^^^10)! * (10^^^^^^^^^10)!^^^^^^(10^^^^^^10)!

Twasbrillig's Up-arrow Salad

                    This is the 17th record setting entry in the "My Number is Bigger" competition. This number is a big jump from the 16th record setter. However it's a salad number. The factorials and multiplication add little to the number and it ends up being not much larger than a Tridecal. This is also the 19th number listed in the forum.

(11^^^^^^^^^^11)! * (11^^^^^^^^^11)!^^^^^^(11^^^^^^11)!

Blatm's Finesse

                    This is the 18th record setting entry in the "My Number is Bigger" competition, and the 20th official entry. This was Blatm's response to Twasbrillig's Up-arrow Salad. We can see that people recognize intuitively that salad numbers are sloppy because Blatm responded by saying "Not one for Finesse, are you?". Blatm simply took Twasbrillig's number and converted to base 11. This makes for a (slightly) larger value, though at this point this likely to be clobbered by whoever is willing to type out more up-arrows, such as a screen fill. As it turns out, this is the last number in the Primitive Recursive Epoch, and the next entry is much much larger!

<10,100,10>

         This would almost certainly be larger than the last two salad numbers. This would be the hypothetical tenth Bowers' elementary googol extension ... which brings us to ...

E100##10

googondol

           My tenth and final elementary googol extension.

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

googondoldecex

           "deca" + "plex" = "decex". This concludes the base operators: plex, dex, threx, tetrex, pentex, hex, heptex, octex, ennex, and decex.

g(1)

2^^^^^^^^^^^^3 = <2,3,12>

                This is the first value in the sequence used to construct Little Graham, the original Graham's Number used in RL Graham's 1971 paper "Ramsey Theory for n-parameter sets".

E100##20

googonkosol

After a googondol I did continue some further extensions, this time jumping by 10s. This name is derived from "googondol" + "icosi"(greek 20). It can be expanded (using Extended Hyper-E rules) to E100#100# ... #100 w/20 100s. This falls in a bit of a deadspot for the numbers, because we are transitioning into higher levels of recursion at this point.

3-->3-->2-->2

3^^^^^^^^^^^^^^^^^^^^^^^^^^^3

                     This is one of the smallest non-trivial cases of Conway's Chain Arrow Notation. Three argument chains are trivial as they are just an alternative to Knuth Arrow notation. 4 element chains are also trivial if any of it's elements are 1 (assuming the other arguments are no larger than say 10). Expanding this expression we get:

 3-->3-->2-->2

3-->3-->(3-->3-->1-->2)-->1

3-->3-->(3-->3-->1)

3-->3-->(3^3)

3-->3-->27

                     That is, we get 3 followed by 27 Up-arrows followed by 3. This places this number somewhere between a tridecal (10-->10-->10) and a boogol (10-->10-->100). It also lies between G(1) and G(2). In Part II we will begin to go through Conway's Chain Arrow Notation in more depth.

E100##30

googontritol

googondol + "trito"(3) for the "30th googol".

E100##40

googonsartol

The 40th googol.

E100##50

googonpetol

The 50th googol.

E100##60

googonextol

The 60th googol.

E100##70

googonheptol

The 70th googol.

E100##80

googonogdol

The 80th googol.

E100##90

googonentol

The 90th googol.

{10,10,100} = <10,10,98>

Boogol Jr.

The original boogol, a googolism coined by Jonathan Bowers. A boogol is 10 hecated to the 10th, or 10{100}10 using Bowers old array notation. However in 2007 Bowers changed the definition of his notation so that <b,p> = b^p instead of the former {b,p} = b+p. In the new notation therefore, {10,10,100} is actually <10,10,98>. That is, two tens between 98 up-arrows. At this scale however this barely seems significantly different than a boogol, and in fact the newer boogol will be the very next entry ...

{10,10,102} = <10,10,100> = 10<100>10

boogol

The post-2007 definition of a boogol, where it is 10^^^...^^^10 w/100 ^s, which honestly ... is pretty cool. This is only "slightly" larger than its original definition which involved being 10 to the 10 under the 100th operation. This boogol is technically 10 (102)-ated to the 10th, which doesn't sound as spiffy. The main reason why the new array notation is preferable, has nothing to do with the numbers being "larger". The "larger" here is so slight in googological terms its barely worth mentioning past this point. Rather it's because of consistency. With <b,p> = b^p we consistently have that <b,1...> = b regardless of whatever follows, since b^1 = b. Notice this is true of all the hyper-operators: b^^1 = b , b^^^1 = b , b^^^^1 = b , etc. With Bowers new array notation this becomes universal and brings great harmony to the whole.

In Hyper-E Notation this boogol can be written as:

E1#1#1# ... #1#1#1#10

w/99 "1#"s

This makes it just ever so slightly less than a gugold. Which brings us to ...

<10,100,100> = 10<100>100

This would be the hypothetical 100th Bowers' googol extension. It could be written as:

E1#1#1# ... #1#1#1#100

w/99 "1#"s

This gets us blisteringly close to a gugold and yet ...

<10,101,100> = 10<100>101

E1#1#...#1#1#10#100 w/98 "1#"s

This number is larger than a boogol but still smaller than a gugold! We can prove this by noting that E1#1#...#1#1#101 = E1#1#...#1#10#100 < E1#1#...#1#100#100 < E100#100#...#100#100#100. And with that we conclude Part I ...

E100##100

gugold

This number is roughly equivalent to 10^^^...^^^100 w/100 ^s = 10<100>100 = <10,100,100>. In actuality it's larger than this. Since Jonathan Bower's boogol = 10<100>10 < 10<100>100 < E100##100, it follows that boogol < gugold.

A gugold is my smallest googolism to use Extended Hyper-E Notation. It can be written in ordinary Hyper-E as:

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

<10,102,100> = 10<100>102

This number will be ever so slightly larger than a gugold, and so acts as a strong upperbound in Array Notation. This number is still vastly smaller than ...

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

The Second stage of the modern Graham's Number. 

g(2)

2<2<12>3>3 

The second stage of the original Graham's Number.

10{10{100}10}10

boogolplex Jr.

The original value of a boogolplex. This is equivalent to 10<(10<98>10)-2>10 using the new array notation. This would be slightly smaller than the modern definition.

10<10<100>10>10

boogolplex

The plex version of a boogol. This value will be slightly smaller than a gugolda-suplex.

E100##100#2

gugolda-suplex

"super" + "plex" = "suplex". This is the first operator beyond the "first row" of operators: plex, dex, threx, tetrex, pentex, hex, heptex, octex, ennex, decex.

10<10<100>102>102

This odd looking number is rather exacting upperbound on a gugolda-suplex. It's very important for all further upperbounds of E# and xE# numbers in array notation, so it bares some explication. We may note that E100#100#...#100#100 < E1#1#...#1#102, regardless of how many 100s there are. We saw some earlier examples of this. It follows that E100##100 < E1#1#...#1#102 w/99 1s. It follows then that E100##(10<100>102) is larger than E100##gugold. We then replace all these 100s with 1 except for the last one which gets replaced with 102, and this can be written sucintly in array notation as 10<10<100>102>102. From here we note that we can turn this into a standard expanded expression by replacing all the numbers with 102. Thus we get 102<102<102>102>102. Which brings us to ...

<102,3,1,2>

From here we can quickly surmise that:

E100##100#n < <102,n+1,1,2>

From here we can create more upperbounds, literally accounting for all the additional boosts from all of those 100s. 

G(3)

3<3<3^^^^3>3>3

g(3)

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

E100##100#3

gugolda-dusuplex 

<102,4,1,2>

A relatively strong upperbound on gugolda-dusuplex in array notation.

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.

{10,100,1,2}

corporal

This is the original Corporal (note the curly braces). This number can be written as:

10{10{ ... 10{10{10}10}10 ... }10}10

w/99 sets of { }

Insane. This number can be bounded in xE# quite easily. First we note that 10{10}10 < 10<10>10 = E1#1#1#1#1#1#1#1#1#10 < E1#1#1#1#1#1#1#1#1#100 < E100##10.

Next we have 10<10<10>10>10. It follows quickly this is less than E100##(E100##10) = E100##10#2. From here we can bound corporal ...

E100##10#99

A nifty upperbound for corporal. What's useful about this is it immediately follows that E100##100#100 is larger. Even crazier ... even this is still smaller than G(100) ...

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. We can also show this is approximately equal to 3^3##4#100, since this is equal to 

3^3##(3^3##4)#99 and 3^3##4 is much much larger than 10, it follows that this is larger than the upperbound just proposed for a corporal ... which brings us to ... 

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>.

<102,101,1,2>

A relatively tight upper-bound on a graatagold.

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!

{10,{10,100,1,2},1,2}

corporalplex

The og corporalplex ... still smaller than G(G(100)) which is still smaller than a graatagolda-sudex ...

G(G(100))

We're way past Yudkowsky's Number but we still are just getting started ...

E100##100#100#2

graatagolda-sudex

"super" + "dex" = "sudex". The continuation from here should be clear.

<102,3,2,2>

A tight upperbound on graatagolda-sudex. Using the previous result we can have a tigher upperbound of <102,<102,101,1,2>+1,1,2> < <102,<102,102,1,2>,1,2> = <102,3,2,2>.

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

{10,10,10,2}

grand tridecal

A Grand Tridecal smaller than E100##100##11, which definitely makes it smaller than a gugolthra. Note that from a corporal to a grand tridecal Bowers skips over all of the exploding operators, passing up a whole row of regiments in my system. This starts to happen more and more as we proceed. Next up is the third row of regiments, beginning with the gugolthra regiment.

E100##100##100

gugolthra

E100##100##100##2

graatagolthra

E100##100##100##3

greegolthra

This number is especially notable as being the last regiment just before Jonathan Bowers' tetratri. An explanation can be found in the following entries to tetratri ...

<102,101,2,3>

A very strong upper-bound on a greegolthra. From here we apply an extended version of the Knuth Arrow Theorem, The Generalized Knuth Arrow Theorem, to obtain an upperbound in base 3:

<102,101,2,3> = 102<2,3>101 < (3<2,3>2)<2,3>101 < 3<2,3>103 = <3,103,2,3>

From here we can observe:

3<2,3>103 < 3<2,3>3<2,3>3 = 3<3,3>3 = <3,3,3,3>

  And thus, we have proven that a greegolthra is smaller than a tetratri.

{3,3,3,3}

tetratri

Bowers' tetratri. The etymology is simple "tetra"(4) + "tri"(3), meaning four 3s in array notation. This number is much much larger than Conway's tetratri, and is larger than a gugolthra

{10,100,3,3}

powplodal

This number is trivially smaller than a tetratri since we've replaced the first two 3s with a 10 and 100 respectively. This however is a lower bound for a grinningolthra. CookieFonster coined the name powplodal for this number, based on the operator {{{3}}} which is called powerexplosion. This is actually a type of number that Bowers' himself skipped over, even though it is arguably a more natural continuation of his googol extensions. A powplodal can be written as 10{{{3}}}100 which follows the general pattern 10@100 for 10 to the 100th under the @th operation.  

E100##100##100##4

grinningolthra

The first regiment larger than Bowers' tetratri. This is proven by the fact that grinningolthra is larger than a powplodal which is trivially larger than a tetratri.

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

This number introduces the 5th root, "peta". This forms the "gugolpeta regiment", or sub-regiment as it were. We can continue as expected with graatagolpeta, greegolpeta, grinningolpeta, etc.

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

gugolhexa

This number introduces the 6th root, "hexa". This forms the "gugolhexa regiment", which contains graatagolhexa, greegolhexa, grinningolhexa, etc. At this point the name construction is routine. A gugolhexa is larger than and comparable to <10,10,100,5> in array notation. <10,10,100,5> is a number Bowers' calls a bigol ... a big ol' number that's for sure :p

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

gugolhepta

Beginning of the gugolhepta regiment. A gugolhepta is larger than and comparable to Bowers' boggol = <10,10,100,6>

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

E100###8

gugolocta

Beginning of the gugolocta regiment. A gugolocta is larger than and comparable to Bowers' bagol = <10,10,100,7>. Bowers jokes to never eat a bagol bagels. This implies that is pronounced /bae-gol/ not /bah-gol/.

E100###9

gugolenna

Beginning of the gugolenna regiment. A gugolenna is larger than and comparable to Bowers' ... well actually Bowers' doesn't have a name for the number this is comparable to, however, it would be comparable to <10,10,100,8>. Actually it would be closer to <10,100,100,8>.

E100###10

gugoldeka

Beginning of the gugoldeka regiment. A gugoldeka is larger than and comparable to <10,10,100,9> in array notation, although its more accurate to compare it to <10,100,100,9>, which it is also larger than. We are now getting close to Bowers' general ... let's proceed more cautiously ...

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

graatagoldeka

At this point the deutero-hyperions are nearly bursting at the seams. In array notation this is larger than but comparable to <10,100,1,10>. A general is <10,10,10,10>. It might seem like that 100 might have something to say about that, but in reality in is vanishingly small. Even <10,3,10,10>  would be vastly larger. So we must keep going ...

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

greegoldeka

Approximately equal to <10,100,2,10>

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

grinningoldeka

Approximately equal to <10,100,3,10>

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

golaagoldeka

Approximately equal to <10,100,4,10>

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

gruelohgoldeka

Approximately equal to <10,100,5,10>

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

gaspgoldeka

Approximately equal to <10,100,6,10>

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

ginorgoldeka

Approximately equal to <10,100,7,10>

E100##100##100##100##100##100##100##100##100##100##9

gargantuuldeka

Approximately equal to <10,100,8,10>

E100##100##100##100##100##100##100##100##100##100##10

googondoldeka

Approximately equal to <10,100,9,10>. This is notable for being the end of the (standard) regiments we can create. This can be thought as occupying position (10,10). This number however can be shown to still be smaller than <10,10,10,10>. Note that, <10,100,9,10> is smaller than <10,<10,10,9,10>,9,10> = <10,3,10,10> is smaller than <10,10,10,10>, which brings us at last to ...

{10,10,10,10}

general / tetradecal

Bowers' first introduces this number as general, following his military naming theme, but then also adds that it may also be called a tetradecal. At this point we have essentially passed up all the standard Saibianisms just before a throogol, but actually there is something even larger before we get there ...

{10,100,10,10}

gecoldeka

Technically larger than a general. The term gecoldeka I'm coining for this number. This is still technically smaller than ... 

E100##100##100##100##100##100##100##100##100##100##11

googoadoldeka

Although not explicitly mentioned, this name is pretty easily expressed following the established naming conventions. A googoadol was introduced as an 11th basic googol extension, and this is a natural continuation of that. This has the distinction of being the first "regiment" to clearly pass up a general, although it's basically in the same ballpark. With that we move onto ...

E100###11

gugolendeka

This is approximately equal to <10,100,100,10> in array notation. Because it's rather trivial to coin. All we need to continue is coming up with one additional root. Let's skip a bit though, to groups of 10 ...

E100###20

gugolicosa

"icosa"(20) in greek. Approximately <10,100,100,19>.

E100###30

gugoltrianta

"trianta" meaning 30 (greek). Approximately <10,100,100,29>.

E100###40

gugolsaranta

"saranta" meaning 40 (greek). Approximately <10,100,100,39>.

E100###50

gugolpeninta

"peninta" meaning 50 (greek). Approximately <10,100,100,49>.

E100###60

gugolexinta

"exinta" meaning 60 (greek). Approximately <10,100,100,59>.

E100###70

gugolebdominta

"ebdominta" meaning 70 (greek). Approximately <10,100,100,69>.

E100###80

gugologdonta

"ogdonta" meaning 80 (greek). Approximately <10,100,100,79>.

E100###90

gugoleneninta

"eneninta" meaning 90 (greek). Approximately <10,100,100,89>. Which at last brings us to ...

<10,100,99,99>

A strong lower-bound on E100###100, which is a throogol. This brings us to ...

E100###100

throogol

Approximately equal to <10,100,99,99>, although slightly larger than this value. It however is much smaller than <10,10,10,100>, which is Jonathan Bowers troogol.

<10,103,99,99>

This is believed to be an upperbound on a throogol. From here it is trivial to show that <10,10,10,100> is larger. We can note that 10<99,99>103 < 10<99,99>10<99,99>10 = 10<100,99>3 << 10<10<10,99>10,99>10 = 10<1,100>3 = <10,3,1,100> < <10,10,1,100> < <10,2,2,100> < <10,10,10,100>.

<10,10,10,100>

troogol

One of Bowers' new googolisms introduced on his 2007 polytope.net website. This value is still vastly smaller than ...

{10,10,10,{10,10,10,10}}

generalplex / tetradecalplex

This number falls between a throogol and a throogola-thruplex. This is because a throogol is approximately {10,10,10,100} where as a throogola-thruplex is approximately {10,10,10,{10,10,10,100}}, making it larger than even the generalplex. 

E100###100#2

throogola-thruplex

Equivalent to to E100###throogol or E100###(E100###100). Originally this is where the operators ended, with this number originally called a grand throogol, thus being the first introduction of the "grand". However I recently came up with using "thru" as a way to continue. "three" + "su(per)" = "thru". It should be obvious this can be continued with: thrudex, thruthrex, thrutetrex, ... , thrusuplex , thrusudex , etc. etc. 

E100###100#100

thrangol

At this point we want to recycle names to avoid as much added complexity as possible. A thrangol is derived from "throogol" + "grangol". By replacing "g" or "gr" with "thr" we can create a whole slew of "throogol" variants of our existing googolisms to continue. Many of these have amusing names that are fun to say. Next up with threagol, then thrigangol, thorgegol, thrulgol, thraspgol, thringorgol, thrargantuul, throogondol, etc.

A thrangol is approximately equal to <10,100,1,1,2>.

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

Approximately <10,100,2,1,2>.

E100###100#100#100#100

thrigangol

Approximately <10,100,3,1,2>.

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

throrgegol

Approximately <10,100,4,1,2>.

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

thrulgol

Approximately <10,100,5,1,2>.

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

thraspgol

Approximately <10,100,6,1,2>.

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

thrinorgol

Approximately <10,100,7,1,2>.

E100###100##100

thrugold

Approximately <10,100,99,1,2>.

E100###100##100##2

thraatagold

Approximately <10,100,1,2,2>

E100###100##100##3

threegold

Approxiately <10,100,2,2,2>

E100###100##100##4

thrinningold

Approximately <10,100,3,2,2>

E100###100##100##5

throlaagold

Approximately <10,100,4,2,2>

E100###100##100##6

thruelohgold

Approximately <10,100,5,2,2>

E100###100##100##7

thraspgold

Approximately <10,100,6,2,2>

E100###100##100##8

thrinorgold

Approximately <10,100,7,2,2>

E100###100##100##100

thrugolthra

Approximately <10,100,99,2,2>

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

thrugoltesla

Approximately <10,100,99,3,2>

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

thrugolpeta

Approximately <10,100,99,4,2>

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

thrugolhexa

Approximately <10,100,99,5,2>

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

thrugolhepta

Approximately <10,100,99,6,2>

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

thrugolocta

Approximately <10,100,99,7,2>

E100###100###100

tristo-throogol

Approximately <10,100,99,99,2>

{3,3,3,3,3}

pentatri

Jonathan Bowers' pentatri. This was among Bowers original set of 121 original googolisms on his 2002 website. It falls somewhere between the large gulf of tristo-throogol and teristo-throogol. It is approximately equal to {10,100,3,3,3} which will be comparable to a saibianism. 

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

teristo-throogol

Approximately <10,100,99,99,3>

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

pesto-throogol

Approximately <10,100,99,99,4>

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

existo-throogol

Approximately <10,100,99,99,5>

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

episto-throogol

Approximately <10,100,99,99,6>

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

ogisto-throogol

Approximately <10,100,99,99,7>

{10,10,10,10,10}

pentadecal

Jonathan Bowers' Pentadecal. We have now passed up most of the Throogol Super-Regiment.

E100####100

teroogol

Approximately <10,100,99,99,99>. On googology wiki this number is approximated to <100,100,100,100,99>.

{10,10,10,10,{10,10,10,10,10}}

pentadecalplex

The plex version of a pentadecal. This falls between a teroogol and a teroogola-teruplex.

E100####100#2

teroogola-teruplex

Approximately but larger than <10,100,99,99,<10,99,99,99,99>>, making it definitely larger than a pentadecalplex.

E100####100###100

teri-throogol

E100####100####100

tristo-teroogol

Approximately <10,100,99,99,99,2>

{3,3,3,3,3,3}

hexatri

One of Jonathan Bowers original 120 named numbers.

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

teristo-teroogol

Approximately <10,100,99,99,99,3>

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

pesto-teroogol

Approximately <10,100,99,99,99,4>

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

existo-teroogol

Approximately <10,100,99,99,99,5>

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

episto-teroogol

Approximately <10,100,99,99,99,6>

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

ogisto-teroogol

Approximately <10,100,99,99,99,7>. This puts it "just below" ...

{10,10,10,10,10,10}

hexadecal

The hexadecal, one of Bowers original 120 named numbers.

E100#####100

petoogol

Approximately <10,100,99,99,99,99>

{10,10,10,10,10,{10,10,10,10,10,10}}

hexadecalplex

One of Bowers original 120 named numbers.

E100######100

ectoogol

Approximately <10,100,99,99,99,99,99>

E100#######100

zettoogol

Approximately <10,100,99,99,99,99,99,99>

E100########100

yottoogol

Approximately <10,100,99,99,99,99,99,99,99>. On googology wiki this number is approximated as <100,100,100,100,100,100,100,100,99>

{10,10(1)2}

{10,10,10,10,10,10,10,10,10,10}

iteral

Jonathan Bowers' iteral, one of his original 120 googolisms. This number passes up everything up to yottoogol.

{3,27(1)2}

{3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}

ultatri

An unusual googolism by Jonathan Bowers. This is the only one to use the "ulta" root, which seems to imply that this is {3,...,3} with {3,3,3} 3s, in Bowers' older array notation. In the older array notation this is equivalent to {3,3,2(1)2}. This implies that ultadecal would correspond to {10,10,2(1)2} which would be a pretty long array indeed.