Consider the following Proposition:

**Proposition:** Let $R$ be a noetherian ring. If $p_0 \subsetneq p_1 \subsetneq p_2$ is a chain of distinct prime ideals in $R$, then there exist infinitely many distinct primes $q$ such that $p_0 \subsetneq q \subsetneq p_2$.

For a proof, see for instance this question. I would like to see a counterexample if we drop the noetherian hypothesis. Should such a ring exists I would find it rather interesting because it would be an example where a "finiteness" hypothesis implies that there are infinitely many of something!