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][1] for discussion and references. 

  [1]: https://mathoverflow.net/questions/136948/dehns-algorithm-for-word-problem-for-surface-groups