Let $F$ be a non-abelian free group and let $G=\prod_\omega F$, the direct product of infinitely many copies of $F$. Then the abelianisation of $G$ has torsion (of order $2$), by a theorem of Kharlampovich and Myasnikov ["Implicit function theorem over free groups and genus problem", Knots, braids, and mapping class groups—papers dedicated to Joan S. Birman].
Granted, this is an example of an unreasonably bad phenomenon, not an unreasonably good phenomenon, but I still couldn't believe my ears when I was told it (by Lars Louder).