I agree with you, there is no way we could compute it with our computers. But there is still number theory with which you can analyze incomprehensibly large numbers and make statements about them. Maybe you could say that the next 10 numbers after googolplex are not prime. That would already be a first step towards an answer. You can't just categorically say it's impossible to find out. It is just that our understanding of mathematics is far from being able to answer this question. Anyway, it's more of a playful question I'm asking just for fun. Whether or not we find an answer is not of importance to me.
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:
