Puzzle: Generate the Highest Bounded Number Using a Limited Number of Characters

A friend and I were sitting in our cubes at work and trying to create the greatest bounded number we could using only a few characters.

We came up with $A(G,G)$, which is the Ackermann function with Graham's number $G$ as the '$M$' and '$N$' variables.

Beyond the fact that this number, though technically a bounded number, seems absolutely unquantifiable, are there larger numbers that we missed?

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Have you seen this: scottaaronson.com/writings/bignumbers.html? – Srivatsan Oct 4 '11 at 20:54
You need to carefully specify what and how many symbols are allowed. For example, I have just defined $H=G!$ and propose $A(H,H)$ as larger. This is in the sense that without context, most would not recognize $A$ and $G$ this way. – Ross Millikan Oct 4 '11 at 21:15
I would specify that characters that preform an operation - such as +, -, *, / would count, and characters that don't preform an operation, such as commas and parentheses, are not included. however, if one was to use a parentheses to preform an operation i.e. a(b), it would be counted. – Christopher Rayl Oct 4 '11 at 21:23
@SrivatsanNarayanan, that was an excellent read. – John Gietzen Oct 4 '11 at 21:27
In the early 1960's, as an undergraduate, I read a math newsletter that had a "Large Number Contest" - define the largest number whose definition could be typed on a postcard. There were many ingenious entries, but the hard part was not defining the entries but comparing them to see which were bigger. That's probably unsolvable. – marty cohen Oct 8 '11 at 23:33