I have a conjecture:
Ultra-Primes-Conjecture. There exists infinitely many primes formed of $2^{F[n!+1]}-1$.
Here $2^p-1$ is Mersenne number, $F[n+2]=F[n+1]+F[n], 0,1,1,2,3,5,8,...$ is Fibonacci number, $n!$ is the factorial function.
$n! + 1$ is prime for (sequence A002981 in the OEIS): $$n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, ...$$ These are $2,3,7,39916801,...$.
Indices of prime Fibonacci numbers are: $$3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757, 35999, 37511, 50833, 81839, 104911, 130021, 148091, 201107, 397379, 433781, 590041, 593689, 604711, 931517, 1049897, 1285607, 1636007, 1803059, 1968721, 2904353, 3244369, 3340367,...$$
So $2^{F[2!+1]}-1 = 3, 2^{F[3!+1]}-1= 2^{13}-1= 8191$ are the only two known Ultra-Primes! I guess after $3$, $8191$ there are more prime heros formed of $2^{F[n!+1]}-1$.
And more, Ultra-Primes increase very quickly, they can completely compare with Graham's Number and TREE(3)
It is not currently known whether there are infinitely many Ultra-Primes. Indeed, we don't even know if there are infinitely many Mersenne primes.
A proof or disproof of your conjecture is out of reach with currently available tools in number theory.