The related problem where only the minimum prime divisor of $x_n$ is added to the list $A_{n-1}$ is open as pointed out by @JoeSilverman. This sequence of primes is called the Euclid-Mullin sequence. See the following links:

https://math.stackexchange.com/questions/272958/are-all-primes-euclid-primes

OEIS link

Now, your sequence has more members than the Euclid-Mullin sequence. Thus it would be at least as hard, if not harder, to prove that some prime is missing in your sequence, given that no one has been able to prove that such a prime is missing in theEuclid-Mullin sequence.

The related problem where only the minimum prime divisor of $x_n$ is added to the list $A_{n-1}$ is open as pointed out by @JoeSilverman. This sequence of primes is called the Euclid-Mullin sequence. See the following links:

https://math.stackexchange.com/questions/272958/are-all-primes-euclid-primes

OEIS link

Now, your sequence has more members than the Euclid-Mullin sequence. Thus it would be at least as hard, if not harder, to prove that some prime is missing in your sequence, given that no one has been able to prove that such a prime is missing in theEuclid-Mullin sequence.

Edit: Thanks to Francois Brunault for pointing out that containment is needed for the above claim to hold. The difficulty of your problem is possibly still similar to Euclid-Mullin. Thanks to Greg Martin for the further reference.