What problems have been frequently computationally verified for large values?

The nonexistence of Fermat primes beyond $65537$ has been verified, as far as I know, to $2^{2^{32}}+1$. Thereby, we have identified the last constructible regular prime-sided polygon up to a point far beyond where any such construction could be carried out. (Based on current theories of quantum gravity, a regular polygon having the shortest possible side length and "only" $2^{2^8}+1$ sides would not fit in the known Universe.)

It was, of course, Euler who first killed Fermat's conjecture that $2^{2^n}+1$ is prime for all natural numbers $n$, by disproving it for $n=5$. Now the opposite conjecture is in vogue, and it has been verified up to $n=32$. Testing of Fermat numbers for primality can be accomplished by Pepin's Test, a stronger form of Fermat's Little Theorem whereby a Fermat number $M\ge 5$ is prime iff $3^{(M-1)/2}\equiv -1 \bmod M$. Because Pepin's test does not directly identify factors when the number is composite, $2^{2^n}+1$ has no known factors, despite being certified composite, for $n=20$ and $n=24$.

See here for a more thorough discussion of Fermat numbers.

Whether or not there exist any odd perfect numbers is an open problem. Numbers up to $10^{1500}$ have been checked (as of $2012$) without any success.