This question has crossed my mind, and I tried finding some solutions to that functional equation, then to find a pattern.
It's surprisingly hard to find real functional equation calculators online, so I only found 3 solutions.
$f(x)=\frac{1}{1-x}=f^{\circ4}$
$f(x)=\frac{1}{2-2x}=f^{\circ5}$
$f(x)=\frac{1}{3-3x}=f^{\circ7}$
I didn't really found more, nor find a pattern, nor anything online. So if anyone has an answer, feel free to share it!