Related to this question, I wonder how often $n!+1$ is a prime?
There is a related OEIS sequence A002981, however, nothing is said if the sequence is finite or not... or anything in that sense...
$n! + 1$ is prime for $n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059, 150209, \dots$, no other factorial primes are known as of May 2014. See here for more info on factorial primes.
Just looking at the heuristics of the problem:
If you pick a random integer $x$, it will be a prime number with a probability about $1 / \ln x$. Now the number $n! + 1$ is not a random integer. We know that $n! + 1$ is not divisible by any prime number $p ≤ n$. A random large integer is not divisible by any prime $p ≤ n$ with probability $(1-1/2)(1-1/3)(1-1/5)...$ which is about $1 / (2 \ln n)$. So the likelihood that $n! + 1$ is a prime is accordingly higher, about $2 \ln n / \ln (n!)$.
Using the Stirling formula, $\ln (n!)$ is about $n \ln n - n$ or $n(\ln n - 1)$. So $n!+1$ is prime with probability about $(2/n)/(1 - 1 / \ln n)$.
The factor $(1 - 1 / \ln n)$ is quite close to 1; the number of primes of the form $n! + 1$ with $n ≤ M$ is about $2 \ln M$. Very roughly agrees with the list of primes given earlier (I think it is a list of known primes, with many numbers in between not examined).
The largest factorial primes are discovered only recently. From an announcement:
PrimeGrid is a set of projects based on distributed computing, and devoted to finding primes satisfying various conditions.
Factorial primes-related recent events in PrimeGrid:
$147855!-1$ found: official announcement
$110059!+1$ found: official announcement
$103040!-1$ found: official announcement
$94550!-1$ found: official announcement
Other current PrimeGrid activities: