Generalised Fortunate Prime Sequences
Husk, 8 bytes
ḟ&gt;1M&lt;¹Π↑

Try it online! (Missing the testcase for perfect numbers because it was too slow, and the one for semiprimes because implementing the list of semiprimes is a challenge itself)
Takes as input S and n, where S is an infinite list.
Explanation
ḟ&gt;1M&lt;¹Π↑
       ↑     Take the first n elements from S
      Π       and get their product
   M ¹       For each element x in S
    &lt;         subtract the product if it is smaller than x, return 0 if it is bigger
ḟ            Find the first element in this list
 &gt;1           that is greater than 1