The following was inspired by Jesse Peterson's answer.

Let $M^n$ be a closed hyperbolic manifold, eg a surface, and consider the action of $\Gamma=\pi_1M$ on the boundary at infinity of hyperbolic $n$-space.  Then each $\gamma\in\Gamma\smallsetminus 1$ has precisely two fixed points, namely the end points of its axis of translation.

Any non-commuting pair of elements whose axes share an end point now provide an example of the kind required.  It isn't too difficult to construct explicit examples.

**UPDATE**

As Torsten Ekedahl points out, the last sentence should have read 'It's impossible to construct such examples'!  In particular, this doesn't work.