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 $x\in  I$.

**Note 1:**  As an interesting aside, for every constant $a\in\mathbb{C}$, there is (unique) formal power series with lowest order term $az$ of the form
$$
f(z) = az+\frac{{a}^{2}}{3!}\,{z}^{3}
+{\frac {{a}^{3} \left( {a}^{2}{+}1 \right)}{5!}}\,{z}^{5}
+{\frac {{a}^{4} \left( {a}^{6}{+}{a}^{4}{+}11\,{a}^{2}{+}1\right)}{7!}}\,{z}^{7}+\cdots
$$
that satisfies $f''(z) = f(f(z))$.  When $|a|<1$, this series converges absolutely and uniformly on the disk $|z|^2\le 6\bigl(1{-}|a|\bigr)$.  (This is likely to be strictly less than the actual radius of convergence, as my method for proving convergence is not sharp.  In fact, my method shows that $|f(z)|\le |z|$ when $|z|^2\le 6\bigl(1{-}|a|\bigr)$, but this is probably not sharp either.)

Though I don't (yet) have a proof, numerical calculations suggest that, when $a$ is a sufficiently small negative real number, the above function $f$ extends real analytically to the entire real line and gives a solution $f:\mathbb{R}\to\mathbb{R}$.

**Note 2:**  More generally, for any two constants $a,b\in\mathbb{C}$, there is a formal power series
$$
f(z) = b+a\,(z{-}b) 
+\frac{b}{2!}\,(z{-}b)^2
+\frac{a^2}{3!}\,(z{-}b)^3
+\frac{ba(a{+}1)}{4!}\,(z{-}b)^4
+\cdots
$$
that has $b$ as a formal fixed point, i.e., $f(b) = b$, so that the composition $f(f(z))$ makes sense as a power series centered at $z = b$
and, formally, $f'(b) = a$, that satisfies $f''(z) = f(f(z))$ as formal power series centered at $z = b$.  Moreover, this is the unique power series centered at $z=b$ that has $f(b) = b$ and $f'(b) = a$ and satisfies $f''(z) = f(f(z))$ as formal power series.  

As in the case $b=0$, when $|a|<1$, so that $f$ is a 'formal contraction' on a neighborhood of $b$, it turns out that the series converges absolutely and uniformly on a disc of the form $|z-b| \le R(a,b)$ for some $R(a,b)>0$, so this gives a two-parameter family of local solutions with a contracting fixed point.  It remains to be seen whether there are values of $(a,b)$ (other than $(0,0)$) for which the corresponding $f$ extends to an entire holomorphic function on $\mathbb{C}$, or even nontrivial values of $(a,b)\in\mathbb{R}^2$ for which $f$ extends analytically to a neighborhood of $\mathbb{R}\subset\mathbb{C}$.