Counterexample for the infinitely many primes between two primes in a Noetherian ring 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!

Consider a non-noetherian valuation ring of rank two. For such an example you can take a look at Examples of Non-Noetherian Valuation Rings.
