According to [Booker - A variant of the Euclid-Mullin sequence containing every prime](https://arxiv.org/abs/1605.08929), 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](https://oeis.org/A000945) contains a decent amount of information. For example, the primes up to 37 do appear within the first 50 terms of the sequence.