TITLE: Is $n! + 1$ often a prime?
QUESTION [45 upvotes]: Related to another question (If $n = 51! +1$, then find number of primes among $n+1,n+2,\ldots, n+50$), I wonder: How often is $n!+1$ a prime?
There is a related OEIS sequence A002981, however, nothing is said if the sequence is finite or not. Does anybody know more about it?

REPLY [10 votes]: Such numbers are called factorial primes. There is only limited number of known such numbers.
The largest factorial primes are discovered only recently. From an announcement of an organization called PrimeGrid PRPNet:

On 30 August 2013, PrimeGrid’s PRPNet found the 2nd largest known
  Factorial prime: $$147855!-1$$ The prime is $700,177$ digits long. The
  discovery was made by Pietari Snow (Lumiukko) of Finland using an
  Intel(R) Core(TM) i7 CPU 940 @ 2.93GHz with 6 GB RAM running Linux.
  This computer took just a little over 69 hours and 37 minutes to
  complete the primality test.

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:

321 Prime Search: searching for mega primes of the form $3·2n±1$.
Cullen-Woodall Search: searching for mega primes of forms $n·2n+1$ and $n·2n−1$.
Extended Sierpinski Problem: helping solve the Extended Sierpinski Problem.
Generalized Fermat Prime Search: searching for megaprimes of the form $b2n+1$.
Prime Sierpinski Project: helping solve the Prime Sierpinski Problem.
Proth Prime Search: searching for primes of the form $k·2n+1$.
Seventeen or Bust: helping to solve the Sierpinski Problem.
Sierpinski/Riesel Base 5: helping to solve the Sierpinski/Riesel Base 5 Problem.
Sophie Germain Prime Search: searching for primes $p$ and $2p+1$.
The Riesel problem: helping to solve the Riesel Problem.