According to Booker - A variant of the Euclid-Mullin sequence containing every prime, as of 2016, this question remains open.
One of the central questions in this area was posed by Mullin [6] in 1963: Does the Euclid–Mullin sequence contain every prime number? Despite a compelling heuristic argument of Shanks [9] that the answer is yes, even the broader question of whether there is any Euclid sequence containing every prime number remains open.
The OEIS contains a decent amount of information. For example, the primes up to 29 do appear within the first 50 terms of the sequence.
