I'll turn YCor's comment (above) into an answer. Surface groups are the simplest (and historically first) non-free example. Let $S_g$ be the surface of genus $g$. Then the fundamental group has presentation
$\pi_1(S_g) \cong \langle a_1, a_2, \ldots, a_{2g-1}, a_{2g} \mid a_1 a_2 \ldots a_{2g} A_1 A_2 \ldots A_{2g} \rangle$.
This presentation is a Dehn presentation. A variant of this fact is due to Dehn. See the following post for discussion and references.
