>2^(2^(2^...-1)-1)-1 seems prime for all nestings, but the numbers get enormous fast

Can you explain this. I read your expression (with infinite nesting) as x=2^(x-1), whose solutions are 1 and 2.

Looking at it as a series, I see: 2-1=1 2^(2-1)-1=1 2^(2^(2-1)-1)-1=1 2^(2^(2^...-1)-1)-1=1

Also, assuming that a sequence does have the properties you describe (always prime, and gets large very fast), it seems like that sequence would either have to be hard to compute, or unknown to all of the mathematicians working on finding large primes.