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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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    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.2011-10-08

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The language has to be specified precisely. Small differences in expressive power translate into giant differences in the size of the numbers that can be named.

A contest in 2001 for largest number generated by a C program of up to 512 characters:

http://djm.cc/bignum-results.txt