Is there a first-order proof on $(\mathbb{N},+,\*,\le)$ that there is no upper bound on primes. ie that $$\neg\exists{q}{\forall{p}{(\forall{m,n}\space m\*n=p\rightarrow m=1\vee n=1)\rightarrow p\le q}}$$

If not, can someone provide an example of an elementary extension of $(\mathbb{N},+,\*,\le)$ in which there is a largest prime.