Find an abelian infinite group such that every proper subgroup is finite
Consider the set of all $2^n$-th roots of unity, as $n$ ranges over the non-negative integers. An infinite subgroup involves elements of arbitrarily high order, which generate everything below them.
More generally, you can show that the abelian groups whose proper subgroups are finite are precisely the Prüfer groups $\mathbb{Z}[p^{\infty}]$.
(Mentioned in Kaplansky's book, Infinite abelian groups, exercice 23.)