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There are no such elements -- the intersection of the derived series of a free group is trivial. In fact, even more is true -- the intersection of the lower central series of a free group is trivial. This is a theorem of Magnus, and by now there are many proofs. The classical one is in the final chapter of Magnus-Karass-Solitar's book on combinatorial group theory.

By the way, a topological proof of this fact (lifting curves to covers to resolve self-intersections, etc) is contained in my paper "On the self-intersections of curves deep in the lower central series of a surface group" with Justin Malestein.

EDIT : I see that you really want finite solvable quotients, not general solvable quotients. It is still true. Fixing a prime $p$, there is a mod $p$ lower central series'' of a group whose quotients are $p$-groups (so finite nilpotent if the group is finitely generated). For a free group, Zassenhaus proved in his paper

H. Zassenhaus, Ein Verfahren, jeder endlichen p-Gruppe eine Lie-Ring mit der Charakteristik p zuzuordnen, Abh. Math. Sem. Hamburg Univ. 13 (1939), 200-207.

that the intersection of the mod $p$ lower central series of a free group is trivial. This can also be deduced from the paper I mentioned with Justin Malestein, at least for the prime $2$ (one of the proofs we give actually yields regular covers whose order is a power of $2$).

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There are no such elements -- the intersection of the derived series of a free group is trivial. In fact, even more is true -- the intersection of the lower central series of a free group is trivial. This is a theorem of Magnus, and by now there are many proofs. The classical one is in the final chapter of Magnus-Karass-Solitar's book on combinatorial group theory.

By the way, a topological proof of this fact (lifting curves to covers to resolve self-intersections, etc) is contained in my paper "On the self-intersections of curves deep in the lower central series of a surface group" with Justin Malestein.