Yes, there are many infinite examples in a paper of Radin and Sadun, some of which were rediscovered in a paper of mine via different methods. For example, the group $$\langle \left( \begin{array}{clcr} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 &0& 1 \end{array} \right), \left( \begin{array}{clcr} 1 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\ 0& \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{array} \right) \rangle $$ is isomorphic to the amalgam $S_{4}{\ast}_{D_{8}}{D_{16}},$ where I am using $D_{m}$ to denote the dihedral group with $m$ elements.

Yes, there are many infinite examples in a paper of Radin and Sadun, some of which were rediscovered in a paper of mine via different methods. For example, the group $$\langle \left( \begin{array}{clcr} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 &0& 1 \end{array} \right), \left( \begin{array}{clcr} 1 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\ 0& \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{array} \right) \rangle $$ is isomorphic to the amalgam $S_{4}{\ast}_{D_{8}}{D_{16}},$ where I am using $D_{m}$ to denote the dihedral group with $m$ elements.

Note that the Euler characteristic ( as extended by Wall) of this group is $\frac{-1}{48},$ whereas the Euler characteristic of the free product $C_{4} {\ast} C_{8}$ is $\frac{-5}{8}.$