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?
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