No matter how long a list you make claiming, "I found all the primes!", if you multiply all the primes on the list and add 1, you always get a new prime not on the list (or a multiple of a new prime). So the primes never run out. Change the number of primes on the list with the slider and check it yourself.
Why does this method work? No matter which prime on the list you divide N = (the product of all primes on the list) + 1 by, the remainder is always 1 (because the original product is exactly divisible by that prime). So N is not divisible by any of the primes on the list.
But every natural number greater than 1 can be factored into primes, so N must be divisible by some prime. That prime cannot be one of the primes on the original list — it must be a new one. Either N itself is a new prime, or there is a new prime that divides N. In the end, the assumption that "we collected all the primes" always breaks down.