Some *really* large numbers that have been featured
in mathematical proofs:

The density of the prime numbers less than
or equal to a number `n` is approximately equal to li(`n`),
which is defined as the integral between 2 and `n` of d`x`/ln `x`.
For relatively small values of `n`, this is an overestimate.
In 1914, J. E. Littlewood proved that, as
you go higher and higher, this formula switches from being an overestimate to
an underestimate and vice versa an infinite number
of times.

In 1933, S. Skewes, a student of Littlewood, proved (although he
did assume the Riemann hypothesis) that the first switch occurs below
`e`^`e`^`e`^79, which is about equal to
10^(10^(10^34)). This number is called Skewes' number.
It is tremendously large, much larger than
the number of particles in the universe or
even a googolplex. Hardy called it "the
largest number which has ever served any definite purpose in mathematics".
Later on he took on the same problem without assuming the Riemann
hypothesis, obtaining the even more enormous answer of about
`e`^`e`^`e`^`e`^67, which is about
equal to 10^10^10^10^29, and in 1955 obtained the "smaller" value of
10^(10^(10^967)).
These numbers are upper bounds, though. Others have since lowered
that bound to "only" 6.69×10^{370}.

The 1933 paper was entitled "On the difference π(x)−li(x), (I)" and was in the Journal of the London Mathematics Society.

Graham's number is listed in the Guinness Book of Records as the highest number ever used in a mathematical proof. This number was derived by R.L. Graham in 1977 from a problem in Ramsay theory (a branch of combinatorics) which concerns bichromatic hypercubes. Normal exponential notation is not enough to express this number, so a special "arrow" notation, which was devised by Knuth in 1976, is used to represent this number.

(If this wasn't what you were looking for, you may also have been looking for the law of large numbers.)