So looking at Euclid's proof he says 1)take a finite family of primes (F) 2)multiply all the primes and add one 3)this new number has at least 1 new prime factor

So I was wondering about what kind of primes you get by recursively feeding this process into it self.

Since the number you must factor grows exponentially, it's hard to get a lot of numerical evidence for what happens.

I calculated a few:

[2]-> [2,3]-> [2,3,7]->[2,3,7,43]->[2,3,7,43,13,139]->[2,3,7,43,13,139,3263443] ->[2,3,7,43,13,139,3263443,547,607,1033,31051]-> cannot factor 113423713055421844361000443

[5] (x5)-> [5,2,3,31,7,19,37,3343,79,193662529] -> cannot factor 234069798025176583891

Obviously quite a few primes are missing, 5,11,19,etc from the first list, but could show up later.

So my question is does a finite family of primes exist that eventually generates all the primes? I figure this probably doesn't have an easy answer, but any information related to this process would be appreciated, or even why it can't be done.