Let $\mu\lll\nu$ be $\sigma$-finite Borel measures, which are not finite, on a topological space $X$. Under what conditions is $0<\operatorname{esssup}(\frac{d\mu}{d\nu})<\infty$?
Hypothesis: Is this only possible when the measures are finite?
Edit: As it was pointed out, the finiteness of $\nu$ and $\mu$ is not enough.
I don't think it has any relation with the measures being finite or not. For instance take $X=\mathbb R$ with the usual topology, $\nu$ the Lebesgue measure, and $\mu=2\nu$. Then $\frac{d\mu}{d\nu}=2$.
You can always look at the problem the opposite way: given a measure $\nu$ and any non-negative $g\in L^\infty(X)$, you can define $$ \mu(E)=\int_E g\,d\nu. $$ If $\nu$ is infinite, $\mu$ will be infinite precisely when $g\not\in L^1(X,\nu)$.