Does there exist a verbally simple group, which is not characteristically simple? A characteristically simple group is a group without non-trivial proper characteristic subgroups, a verbally simple group is a group without non-trivial proper verbal subgroups.
If such group exists, it has to be infinite, as every finite group is characteristically simple iff it is verbally simple (the proof of this fact can be found here: Does there exist some sort of classification of finite verbally simple groups?). However that proof, relies strongly on mathematical induction by the group order and is thus valid only for finite groups. And I do not know whether such infinite group exists.