Note: This is based on Aarex's

First, let G(0) = 4 and G(n) = 3^(G(n-1))^3

Graham's number is G(64)

Let GP(0) = G(64) and GP(n) = G(GP(n-1))

Graham Power Number is GP(64)

Let GT(0) = GP(64) and GT(n) = GP(GT(n-1))

Graham Tower Number is GT(64)

Let GS(0) = GT(64) and GS(n) = GT(GS(n-1))

Graham Stack Number is GS(64)

Let GD(0) = GS(64) and GD(n) = GS(GD(n-1))

Graham Diamond Number is GD(64)

Let G(n) = 1G(n), GP(n) = 2G(n), GT(n) = 3G(n), GS(n) = 4G(n), GD(n) = 5G(n), etc.

Graham Super Number is 64G(64)

Let G1(n) = G(n)

Let G2(0) = 64G(64), and G2(n) = (G1(n-1))G(G1(n-1))

Super Graham Number is 2G(64)

Let G1(n) = 1G2(n)

Let GP1(n) = 2G2(n)

Let 2G2(0) = 1G2(64), and 2G2(n) = 1G2(2G2(n-1))

Super Graham Power Number is 2G2(64)

Super Graham Super Number is 64G2(64)

Ultimate Graham Number is 64G64(64)

Aarex continued:

Ultimate Graham Number = G2(0)

G2(1) = Ultimate Graham Number, replacing 64 into G2(0)

G2(2) = Ultimate Graham Number, replacing 64 into G2(1)

2G2(0) = G2(64) = Ultimate Graham Power Number

3G2(0) = 2G2(64) = Ultimate Graham Tower Number

G22(0) = 64G2(64) = Ultimate Super Graham Number

G23(0) = 64G22(64) = Ultimate Super Super Graham Number

G3(0) = 64G264(64) = Ultimate Ultimate Graham Number

G4(0) = 64G364(64) = Ultimate Ultimate Ultimate Graham Number

64G6464(64) = Ultra Graham Number

I continued:

GG(0) = Ultra Graham Number

GG(1) = Ultra Graham Number, replacing 64 into GG(0)

GG(64) = Ultra Graham Power Number

2GG(64) = Ultra Graham Tower Number

GG2(64) = Ultra Super Graham Number

GG_{2}(64) = Ultra Ultimate Graham Number

GGG(0) = 64GG_{64}64(64) = Ultra Ultra Graham Number

64GGG...GGG64_{64}(64) with 64 G's = Hyper Graham Number

Aarex continued:

G((0)) = Hyper Graham Number

G((1)) = Hyper Graham Number, replacing 64 into G((0))

2G((0)) = G((64)) = Hyper Graham Power Number

G2((0)) = 64G((64)) = Hyper Super Graham Number

G2((0)) = 64G64((64)) = Hyper Ultimate Graham Number

GG((0)) = 64G6464((64)) = Hyper Ultra Graham Number

G(((0))) = 64GGG...GGG6464((64)) with 64 G's = Hyper Hyper Graham Number

G((((0)))) = 64GGG...GGG6464(((64))) with 64 G's = Hyper Hyper Hyper Graham Number

64GGG...GGG6464(((...(((64)))...))) with 64 G's and 64 ()'s inside out = Godly Graham Number

I continued:

G{0} = Godly Graham Number

G{1} = Godly Graham Number, with 64 replaced to G{0}

2G{0} = Godly Graham Tower Number

G2{0} = Godly Super Graham Number

G_2{0} = Godly Ultimate Graham Number

GG{0} = Godly Ultra Graham Number

G{(0)} = Godly Hyper Graham Number

G{{0}} = Godly Godly Graham Number

64GGG...GGG_(64)64{{{...{{{(((...(((64)))...)))}}}...}}} with 64 G's and {} and () levels = Omnipotent Graham Number

Let replace {} into (0)2.

G(0)_3 = Omnipotent Graham Number

G(1)_3 = Omnipotent Graham Number, with 64 changed to G(0)_3

2G(0)_3 = Omnipotent Graham Tower Number

G2(0)_3 = Omnipotent Super Graham Number

G_2(0)_3 = Omnipotent Ultimate Graham Number

GG(0)_3 = Omnipotent Ultra Graham Number

G((0))_3 = Omnipotent Hyper Graham Number

G{0}_3 = Omnipotent Godly Graham Number

G((0)_3)_3 = Omnipotent Omnipotent Graham Number

G(0)_4 = Fourth Unspeakable Graham Number

64GGG...GGG_(64)64(((...((( ... (((...(((64)))...))) ...)_64)_64)...)_64)_64)_64 with 64 G's, parenthesis operators levels and parenthesis operators = Unbeliavable Graham Number

G(0)-1 = Unbeliavable Graham Number

G(1)-1 = Unbeliavable Graham Number, but with 64 changed to G(0)-1

G2(0)-1 = Unbelievable Graham Tower Number

2G(0)-1 = Unbelievable Super Graham Number

G_2(0)-1 = Unbelievable Ultimate Graham Number

GG(0)-1 = Unbelievable Ultra Graham Number

G((0))-1 = Unbelievable Hyper Graham Number

G(0)_2-1 = Unbelievable Godly Graham Number

G(0)_3-1 = Unbelievable Omnipotent Graham Number

G(0)-1-1 = Unbelievable Unbelievable Graham Number

G(0)-2 = Second Unbelievable Graham Number

G(0)--1 = First-Second Unbelievable Graham Number

G(0)--2 = Second-Second Unbelievable Graham Number

G(0)---1 = First-Third Unbelievable Graham Number

Number made with the Unbelievable Graham Function with 64 nests of anything = Swag Graham Number

New extension

Multivariable Graham Function:

G(n,1) = G(G(G(...(G(G(n)))...))) with n levels

G(n,m) = G(G(G(...(G(G(n,m-1),m-1),m-1)...),m-1),m-1),m-1) with n levels

G(n,0) = G(n)

G(n,m,x) = G(G(G(...(G(G(G(n,m-1,x),m-1,x),m-1,x)...),m-1,x),m-1,x),m-1,x) with n levels

G(n,0,x) = G(n,G(n,G(n,(...(G(n,G(n,G(n,n,x-1),x-1),x-1))...),x-1),x-1),x-1) with n levels


(sorry, don't know how to put this in a ruleset)

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