I gave an example [here][1] of a topological proof that a pair of commutators in a free group is not itself always a commutator.  In answer to the same question, Arturo Magidin indicated how to give a combinatorial proof.  I think it's fair to say that the two proofs have completely different flavours, although you can judge for yourself which is 'simpler'.


  [1]: https://mathoverflow.net/questions/44269/commutator-subgroup-does-not-consist-only-of-commutators/74345#74345