Intersection of conjugate subgroups with infinite index.

Is there a group $G$ with a subgroup $H\subseteq G$ of finite index and an element $g\in G$ such that $H\cap g^{-1} H g$ has infinite index in $G$?

This is not possible, since $[G: H \cap K] \leq [G:H] \cdot [G:K]$ if $H$ and $K$ are subgroups of $G$.