This is a twist on Aarex's Graham Generator, that I think might be a little faster. I have been working on this on paper for the past 2 nights, and it was tiring, but pretty fun.

Sponge's Graham Generator

  • \(G_{n(1)}\) is defined as \(G_{G_{n}}\)
  • \(G_{n(2)}\) is defined as \(\underset{G_{n(1)}}{\underbrace{G_{G_{G_{G_{._{._{._{G_{n}}}}}}}}}}\)
  • \(G_{n(3)}\) is defined as \(\underset{G_{n(2)}}{\underset{\underbrace{G_{G_{G_{G_{._{._{._{G_{n}}}}}}}}}}{\underset{.}{\underset{.}{\underset{.}{\underset{\underbrace{G_{G_{G_{G_{._{._{._{G_{n}}}}}}}}}}{\underset{\underbrace{G_{G_{G_{G_{._{._{._{G_{n}}}}}}}}}}{\underbrace{G_{G_{G_{G_{._{._{._{G_{n}}}}}}}}}}}}}}}}\) with \(G_{n(2)}\) lines of \({G_{G_{G_{G_{._{._{._{G_{n}}}}}}}}}\)'s.
  • \(G_{n(4)}\) is defined as Googological Notation - Sponge's Graham Generator - Representation 3 (v2)
  • \(G_{n(m)}\); m representing the amount of recursive procedures minus one, with each ending amount being \(G_{n(m-1)}\). That was a bit hard to explain, so review the pattern of \(G_{n(1)}\) to \(G_{n(4)}\) to get the idea.

Multiple Parentheses


  • \(G_{n((0))}\) = \(G_{n}\)
  • \(G_{n((1))}\) = \(G_{n(G_{n})}\) = \(G_{n(G_{n((0))})}\)
  • \(G_{n((2))}\) = \(G_{n(G_{n(G_{n})})}\) = \(G_{n(G_{n((1))})}\)
  • \(G_{n((3))}\) = \(G_{n(G_{n(G_{n(G_n)})})}\) = \(G_{n(G_{n((2))})}\)
  • \(G_{n((m))}\) = \(G_{n(G_{n((m-1))})}\)


  • Uniket-uni graham or Great graham = \(G_{64(1)}\) or \(G_{G_{64}}\)
  • Uniket-duo graham or Great great graham = \(G_{64(2)}\)
  • Uniket-tria graham or Great great great graham or 3-ex-great graham = \(G_{64(3)}\)

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