I was on Reddit earlier and answered a question about the usual proof that there are infinitely many primes: multiply any finite set of them, add 1, factor, and you get factors that are not in the set.
But a question occurred to me that I've never seen discussed, and I'm curious if any work has been done or if a conclusive answer has been reached. The question relates to the following sequence. Define:
$S_0 = \{2\}$
$S_n = S_{n-1}\cup T_{n-1}$
where $T_{n-1}$ is defined as the set of prime factors of $\Pi S_{n-1}+1$ (is there a standard notation for "the set of prime factors of n?"). The question is, do all primes eventually appear in one of the $S_n$?
Background: This is closely related to the Euclid-Mullin sequence, and this other version, which are the smallest and largest prime factors of $\prod p_i+1$, respectively.
A complete answer to the question posed can be found in the following paper (link):
A. R. Booker, On Mullin's second sequence of primes, Integers, 12A (2012), article A4.
Specifically, see p.3, Variant #2. This is precisely the question of OP, and the answer is "a very thin subset of the primes", because each prime generated divides a Sylvester number, and the latter were proven to be sparse in the following paper, for which I do not have a link.
R. W. K. Odoni. On the prime divisors of the sequence $w_{n+1} = 1 + w_1 \cdots w_n$. J. London Math. Soc. (2), 32(1):1–11, 1985.