I understand that it is an open problem whether there are an infinite number of composite numbers of the form $2^p-1$ with $p$ prime.

Is it possible to find examples of such numbers that are much larger than the largest known Mersenne prime? What is the largest known? I was thinking that short of solving the open problem I could try to show that $2^{n^2+1}-1$ is prime for only finitely many $n$, since it is also unknown if there are infinitely many $n$ with $n^2+1$ prime. Is there any conjecturally infinite set of primes that is easily computable for which $2^p-1$ is always composite?