What are some interesting sequences that contain infinitely many primes?

If it takes form of a polynomial, Dirichlet's theorem answer the question completely for linear polynomial. What about polynomials of degree more than 1? Is there a known polynomial of degree more than 1 that contains infinitely many primes?

What about more complicated sequences like $2^n+3^n$, $n!+1$, etc?

Please provide examples that are as interesting as possible, accompanied with proofs (or reference to proofs) if not too difficult.

Thanks in advanced.