Such numbers are best understood in terms of some computability hierarchy. A common starting point is Ackermann's function, a version of which grows faster than any primitive recursive function (i.e. any fortran function definable with for loops, successor, and unlimited resources) . I will let you look up the function so that I do not misstate its definition. A(4,4) is something like the number of quarks in the universe, A(5,5) the number of arrangements of such quarks, and Graham's number is something like A(6,6). (I am making these up. Check elsewhere for the actual numbers. The above is just to give you an idea of scale.)

EDIT: Another poster has fixed the idea of scale, showing that Graham's number is much larger than I represent above. That poster also goes on to mention n(4) from the article mentioned below is in turn much larger than Graham's number which is much larger than n(3).
END EDIT

However, Graham's number is near the beginning of a list of enormous numbers. Harvey Friedman has a paper on some nice combinatorial problems whose answers go far beyond Graham's number. One of them is a simple sequence problem which starts out something like 2, 12, B, where B is somewhere near A(300,300). If more details come to mind, I will post them here.

In fact, I will post a section from Friedman's paper, 'Enormous Integers in Real Life'. He asked several people to guess at the size of B, which in the quote below is n(3), including L. Lovasz, given the description of the sequence. Lovasz guessed 20,000; Friedman uses A(n) instead of A(n,n):

"THEOREM 8.2. n(3) > A(7,184).

Lovasz wins, as his guess is closer to A(7,184) than the
other guesses. "

Gerhard "Ask Me About System Design" Paseman, 2010.01.15

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