I know that's probably not what you mean but there is money prizes: https://www.eff.org/awards/coop

Because they're there...

There's no cryptographic use in finding a largest prime. It's far larger than any prime used in a practical system and doesn't contribute any mathematical knowledge about primes.

It's mostly just running an algorithm for long enough to pay off.

Come on, stop spreading nonsense. These huge primes are irrelevant for cryptography.