There are infinitely many prime numbers. This was proven by Euclid over 2000 years ago.
Proof sketch: assume there are finitely many primes. Multiply them all together and add 1. This new number is not divisible by any of the primes (remainder 1). So it is either prime itself, or divisible by a new prime. Contradiction.
As numbers grow, primes become rarer. But they never run out.