I used to do very large computations of pi, and even wrote several programs to do it. I see this as analogous to computing (a) the primality of large numbers, and (b) finding very large primes. I computed pi because, well, it was fun! There was no practical reason to do so. Sure, I could do some statistical tests on the digits or search for my phone number and date of birth in the string of digits, but otherwise it was just an endeavor in pushing my hardware to its "computational limits" and combining what I know about programming and mathematics to make some integrated product (no pun intended).
So, while I can't speak for the prime number crunchers, I believe it is just done "just because". It also gives an avenue to try new and more efficient numerical algorithms for multiplication and related operations, since the programs that compute large primes use these algorithms extensively.
Edit: In response to the part about there being an infinite number of primes, while there certainly are an infinite number, generating the *N*th prime isn't a trivial task (while, for example, computing pi to a million decimal places is a trivial task). There are "prime number formulas" that can give one the *N*th prime directly and deterministically, but these algorithms/formulas are incredibly inefficient, usually derived from Wilson's theorem. Mill's theorem provides another prime number formula, but requires the value of a certain constant called "Mill's constant". But to find that constant, you must find primes beforehand (this doesn't make Mill's theorem irrelevant, just not useful for the computation of primes).