this is totally elementary, but I have no idea how to solve it: let $A$ be an abelian group such that $A$ is isomorphic to $A^3$. is then $A$ isomorphic to $A^2$? probably no, but how construct a counterexample? you can also ask this in other categories as well, for example rings. if you restrict to boolean rings, the question becomes a topological one which makes you think about fractals: let $X$ be stone space such that $X \cong X + X + X$, does it follow that $X \cong X + X$ (here + means disjoint union)?
edit: in the answers there are already counterexamples. but you may add others in other categories (with products/coproducts), especially if they are easy to understand :).