You can decide whether a given element in a torsion-free hyperbolic group is a square. This class includes finitely generated free groups, torsion-free groups acting (nicely) on trees, and fundamental groups of closed surfaces with negative Euler characteristic.
Suppose $\Gamma=\langle a_1,\ldots,a_k\rangle$ and $a_0\in\Gamma$ is the element you're interested in. You can say $\Gamma=\langle a_0,a_1,\ldots,a_k\rangle$, so then the validity of following sentence can be decided by [Sela, §4]: $$\exists x\ (a_0=x^2)$$ Technically this should begin with $\forall y$ to make an AE sentence, but the AE sentence $$\forall y\exists x\ (y=y)\wedge(a_0=x^2)$$ is equivalent.