It's a perfectly fun and good question, I just don't think the answer is that hard. Short of a truly titanic mathematical breakthrough, it's not possible to find out. Beside this being one of the oldest, most worked class of problems in mathematics, the people doing these large and systematic searches are not ignorant of number theory:
While you can easily tell some numbers around googolplex are not prime (googolplex + 1 and googolplex + 2 are obviously not) this doesn't tell you anything useful about where the next prime might be. Testing a single such gigantic number for primality is infeasible and there aren't exactly a lot of prime numbers out there - on average, one every couple of googols as you can see from:
https://en.wikipedia.org/wiki/Great_Internet_Mersenne_Prime_...
https://en.wikipedia.org/wiki/Repunit#Repunit_primes
While you can easily tell some numbers around googolplex are not prime (googolplex + 1 and googolplex + 2 are obviously not) this doesn't tell you anything useful about where the next prime might be. Testing a single such gigantic number for primality is infeasible and there aren't exactly a lot of prime numbers out there - on average, one every couple of googols as you can see from:
https://en.wikipedia.org/wiki/Prime_number_theorem
Even if you could test them very quickly, you'd be looking for quite a while.