Factorial primes are are primes of the form $n! \pm 1$ and primorial primes are primes of the form $p\#\pm 1$, where $p\#$ is the product of all primes $\leq p$.
To cite http://www.ams.org/journals/mcom/2002-71-237/S0025-5718-01-01315-1/: "Careful checks over the last half-century have turned up relatively few such primes". However, they also made the following two conjectures:
The expected numbers of factorial primes of each of the forms $n! \pm 1$ with $n \leq N$ are both approximately $e^\gamma \log N$.
The expected numbers of primorial primes of each of the forms $p\# \pm 1$ with $p \leq N$ are both approximately $e^\gamma \log N$.
I'm somewhat curious about twin primes of the forms $(n!-1,n!+1)$ and $(p\#-1,p\#+1)$. If the conjectures are right, we could ask ourselves if there are infinitely many twin primes of these forms. But clearly, as neither these conjectures are proven and neither is the twin prime conjecture, a positive answer is impossible.
However, what about the converse? Is it possible to prove there can only be finitely many twin primes of these form(s)?
Note: twin primes of both forms exist: $$(3!-1,3!+1) = (5,7)$$ $$(3\#-1,3\#+1) = (5,7)$$ $$(5\#-1,5\#+1) = (29,31)$$ $$(11\#-1,11\#+1) = (2309,2311)$$ There are the only ones known (so far).