Suppose $P,Q$ are two probability distributions on a measurable space $(X, \mathcal F)$. The KL-divergence between $P$ and $Q$ is defined as
\begin{equation} D_{KL}[P||Q] = \int_X \log \frac {dP}{dQ}dP \end{equation}
when $P$ is absolutely continuous wrt $Q$, and $ + \infty$ otherwise. So we know that $P$ not absolutely continuous wrt $Q$ yields $D_{KL}[P\|Q]=+\infty$.
My question is whether the converse holds: are there distributions with $P \ll Q$ (absolutely continuous) and $D_{KL}[P\|Q]=+\infty$?
Yes, such distributions exist. Let $P, Q$ be distributions on $\mathbb N$ such that for each $n > 0$ we have $P(n) = 2^{-n}$, and let $Q(n) = 2^{-2^n-n}$, with $Q(0)$ chosen so that the total probability of $Q$ equals 1. Then $P \ll Q$, but $$ D_{KL}[P\mid\mid Q] = \sum_{n \in \mathbb N} P(n) \log\left(\frac{P(n)}{Q(n)}\right) = \sum_{n>0} 2^{-n} \log\left(\frac{2^{-n}}{2^{-2^n-n}}\right) = \sum_{n > 0} 1 = \infty. $$