If $f$ and $g$ are both linear, it's not difficult to find examples, e.g. $f(x)=7x-3$ and $g(x)=11x-5$; in general, for $f(x)=ax+b$ and $g(x)=cx+d$, the necessary and sufficient condition is easily seen to be $\frac{a-1}{c-1}=\frac{b}{d}$.
For non-linear examples, however, I've only been able to come up with relatively "trivial" examples, like $f(x)=\sqrt[3]{x}$ and $g(x)=x^{6}$, or $f(x)=-x$ and $g(x)=\frac{1}{x}$.
So my two-part question is:
(a) Are there any necessary or sufficient conditions so that for non-inverse, non-equal $f$ and $g$, $f\circ g = g\circ f$?
(b) Are there more complex examples than the above for which at least one of the two functions is non-linear? If so, I'd love any insight into how they were found.