Actually, there are lots of *local* solutions of this equation.  For example, suppose that we start with a $C^2$ function $f$ on an interval $I\subset\mathbb{R}$ such that $f'$ is positive on $I$ and $f(I)$ is disjoint from $I$.  Then an inverse $g:f(I)\to I$ of $f:I\to f(I)$ exists and is $C^2$.  Now define $f$ on the interval $f(I)$ so that $f(y) = f''(g(y))$ for $y\in f(I)$.  Then for $x\in I$, we will have $x = g(y)$ for some $y\in f(I)$ and, of course, $y = f(x)$.  Then $f''(x) = f''(g(y)) = f(y) = f(f(x))$ for all $x\in  I$.