PART 4 (10{100}10 ~ {10, 100(1) 2})
The Expansion Range (10{100}10 ~ 10{{1}}10100)
200{200}199{200}198{200}197...4{200}3{200}2{200}1
The hyperfaxul.
~ 10{mega-3}(10{mega-3}2)
Moser's number, equal to 2 in a polygon where the number of sides is the already gigantic mega. The last 8 digits are ...80301056.
3{3{4}3}3
The second term in the sequence defining Graham's number.
(27{25}27){27{25}27 - 2}(27{25}27)
The troogathree, or booga applied 3 times to 3.
3{3{3{4}3}3}3
The third term in the sequence defining Graham's number.
2{2{2{2{2{2{2{12}3}3}3}3}3}3}3
Little Graham, the number that was actually the upper bound to the solution of the problem from which Graham's number arose.
3{3{3{3{3{3{3{3{4}3}3}3}3}3}3}3}3
G8, the first term in Graham's sequence larger than Little Graham.
G9
G10
Troogaten
Comparable to 10{10{10{10{10{10{10{10{10{10{8}10}10}10}10}10}10}10}10}10}10.
fw+1(10)
The number that Denis Maksudov calls unaddom. It is equal to fw(fw(fw(fw(fw(fw(fw(fw(fw(f10(10)))))))))).
G11
G12
G13
G14
G15
G16
G17
G18
G19
G20
G21
G22
G23
G24
G25
G26
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{27}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 -> 3 -> 27 -> 2 in Conway's chained arrow notation. The number of times you would need to iterate the number of arrows to reach Conway's tetratri (which we will encounter later).
G54
G55
G56
G57
G58
G59
G60
G61
G62
G63
3{{1}}65
A BEAF lower-bound to Graham's number, starting with 3^^^3 instead of 3^^^^3.
g(64, 1, 1, 5, 3, 3)
A lower-bound to Graham's number using Joyce's g-function. In fact, if the g-function were defined so that g(a, b, c) was equal to c^ab instead of c^a-1b, then g(64, 1, 1, 4, 3, 3) would be exactly equal to Graham's number.
G64
Main article: Graham's number
Graham's number, the famous number that was devised as an upper bound to the solution of a problem in Ramsey theory.
10{{1}}100
The corporal, the first of Bowers' googolisms that is greater than Graham's number. It is reached by starting with 10^^^^^^^^^^10 (the tridecal) and iterating the number of arrows 99 times.
~ 9{{1}}116
The output of ioannis.c, or Ioannis' number. The last 8 digits are ...82688009.
G1,000,000
The forcal.
The Chained Arrow Range (10{{1}}10100 ~ {10, 10, 10, 100}
trooga(googol)
The bed. Trooga means booga applied n times to n (we saw some values of this earlier in the list).
fw+1(f8(8))
fw+2(2) in the fast-growing hierarchy. This is also equal to fw2(2), fw^2(2), fw^w(2), fe0(2), and (under my interpretation) fζ0(2), fη0(2), fphi(4, 0)(2), fphi(w, 0)(2), and even fΓ0(2). The last 4 digits are ...1248.
3 -> 3 -> 3 -> 3
Conway's tetratri. This number can be approximated by fw+1(fw+1(27)) in the fast-growing hierarchy, and is reached by iterating the number of up-arrows in the same way used to define Graham's number, only starting with 27 instead of 3^^^^3, and with (3 -> 3 -> 2 -> 3) - 1 iterations instead of just 64.
fw+2(10)
The number that Denis Maksudov named baddom.
4 -> 4 -> 4 -> 4
Conway's tetratet, the largest number mentioned in the Book of Numbers. It can be approximated by fw+2(fw+2(fw+2(256))).
10 -> 10 -> 10 -> 10
{3, 3, 3, 3}
Bowers' tetratri.
5 -> 5 -> 5 -> 5 -> 5
The fifth Conway number. It is approximately fw+3(fw+3(fw+3(fw+3(3125)))).
{4, 4, 4, 4}
Bowers' supertet.
{10, 10, 100, 4}
{10, 10, 100, 5}
{10, 10, 100, 6}
The boggol. Because the name is pronounced the same as "boggle", Bowers said of this number, "Boggles the mind, doesn't it?".
{10, 10, 100, 7}
The bagol. Bowers jokingly said "whatever you do, don't eat this many bagels".
{10, 10, 10, 10}
The general, or alternatively tetradecal.
The Linear Array Range ({10, 10, 10, 100} ~ {10, 100(1) 2})
{3, 3, 3, 3, 3}
The pentatri.
{7, 7, 7, 7, 7, 7, 7}
The supersept.
{8, 8, 8, 8, 8, 8, 8, 8}
The superoct.
{3, 3, 3, 3, 3, 3, 3, 3, 3, 3}
{10, 10, 10, 10, 10, 10, 10, 10, 10, 10}
The iteral.
fw^w(10)
The number that Denis Maksudov calls dekexom. This entry is used to discuss the order type ww in the fast-growing hierarchy, which represents the growth rate of Bowers' linear arrays, and the first part of my extension of Steinhaus-Moser notation (which is really just an offset form of Hyper-Moser Notation).
{3, 27(1)2}
Bowers' ultatri. It is a linear array of 27 threes.