There is no largest prime. In fact the classic proof that there are infinitely many primes starts with the assumption that there is a largest prime and proceeds to use this number to construct a larger number which is also prime. This is a contradiction: "this is the largest prime" and "this is not the largest prime." So the original assumption must be false and "there is no largest prime" or equivalently "there are infinitely many primes". Recently it has been shown that this proof is a result of misunderstanding the one given in Euclid's Elements. Nevertheless it's still a fine proof.