EDIT: I forgot how open number theory is! (I think that gets me put on mathematician probation or something.) For this question, I will accept any answer which assumes "standard conjectures" such as RH, Schanuel, Bunyakovsky, etc. I'm definitely more optimistic about conditional results.
Say that a factonomial sequence is a (possibly infinite) sequence of natural numbers $x_i$ such that
each $x_i$ is prime, and
there is some (single variable, integer coefficients, nonconstant) polynomial $p$ such that for all $i>1$ we have $$x_i=p(\prod_{j<i}x_j).$$
Call the polynomial $p$ the shape of the factonomial sequence; each factonomial sequence is determined by its shape and its initial value.
The basic idea is that factonomial sequences come out of some elementary proofs that infinitely many primes of a certain form exist. For instance, the usual "multiply everything and add one" argument gives rise to the shape $p_1(u)=u+1$, and the usual "multiply everything twice and add two" proof that there are infinitely many primes $\equiv 3 (\mathsf{mod}$ $4)$ gives rise to the shape $p_2(u)=u^2+2$. The maximal factonomial sequence with shape $p_2$ and starting value $3$ is $$3,11,1091, 1296216011, 2177870960662059587828905091.$$ The next term would be $$10329907495268194677701503661780370732730049826819138974714891651071966324541232011,$$ but that's not prime. (Amusingly, its smallest prime factor is $41$, but its smallest prime factor which is $3$ mod $4$ is a bit bigger: $76870667$.)
Question: is there an infinite factonomial sequence?
(This is a question which I'm 99% sure one of my students will ask me in the next couple days!) I'm aware of multiple results of the form "no function of such-and-such type has output consisting entirely of primes," but I don't immediately see one which applies here. Unfortunately, terms in factonomial sequences grow so fast that I can't do much experimenting.