Question. Is there a nontrivial group $G$ such that $[G,G]=G$ and $g^3=1$ for every $g\in G$?
All I could think of so far is the following.
If such a group exists it must be infinitely generated due to the local finiteness of groups of exponent 3.
Perfect groups of large enough prime exponent do exist, e.g. a Tarski Monster.