Infinitely many primes
JavaScript (ES6),  45  44 bytes
Takes input as (n)(p1), where \$n\$ is 0-indexed.

n=&gt;g=(p,d=2)=&gt;n?~p%d?g(p,d+1):--n?g(p*d):d:p

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Commented
n =&gt;                // n = 0-based index of the requested term
  g = (             // g is a recursive function taking:
    p,              //   p = current prime product
    d = 2           //   d = current divisor
  ) =&gt;              //
    n ?             // if n is not equal to 0:
      ~p % d ?      //   if d is not a divisor of ~p (i.e. not a divisor of p + 1):
        g(p, d + 1) //     increment d until it is
      :             //   else:
        --n ?       //     decrement n; if it's still not equal to 0:
          g(p * d)  //       do a recursive call with the updated prime product
        :           //     else:
          d         //       stop recursion and return d
    :               // else:
      p             //   don't do any recursion and return p right away