Is it known whether there is any example of a pair of rotations in $SO(3)$ about orthogonal axes such that the group that they generate is not a free product of the two cyclic groups generated by each generator?



finitely generated subgroups of SO(3)
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 $\frac{-5}{8}.$


