I am looking for simple examples of finitely generated residually finite group $G$ with a subgroup of finite index $H<G$ isomorphic to $G$. There are virtually nilpotent groups with this property, lamplighter groups $\mathbb{Z}_p\wr\mathbb{Z}$, Baumslag-Solitar groups. I am interested in "essentially different" examples.

Is there a just-infinite group $G$ with a subgroup of finite index isomorphic to $G$ and which is not virtually nilpotent?