Are there any integral domains in which no nonzero prime ideal is finitely generated? (Other than fields, of course, where the condition is vacuously satisfied.)
I asked a similar question the other day, but the solution there relied on using zero-divisors and that didn't really help clear up the situation I was considering.
Every valuation ring of rank one which is not discrete satisfy your requirement.
A concrete example you can find here. Another one is the integral closure of $\mathbb Z_p$ (the ring of $p$-adic integers) in $\overline{\mathbb Q}_p$ (the algebraic closure of $\mathbb Q_p$, the field of $p$-adic numbers).