Senior Kangaroo 2017 Q19

The sequence of functions $F_1\left(x\right)$, $F_2\left(x\right)$, ... satisfies the following conditions:

$F_1\left(x\right)=x$, $F_{n+1}\left(x\right)=\frac{1}{1-F_n\left(x\right)}$

The integer $C$ is a three-digit cube such that $F_C\left(C\right)=C$

What is the largest possible value of $C$?