Hey future people. Just necroing this to point out the best answer, which we missed. Fermat primes, of which the highest known is currently 65537. It's a very interesting set, because they determine which shapes can be constructed by compass and straightedge, which might be the oldest big question in recorded mathematics.

Like the Wiefrich primes, it's possible there's more, and unlike them it's not, as far as I can tell, widely thought they are finite (it's more up in the air). However, I think the interestingness outweighs that.