I think there can't be a positive radius of convergence.
I don't know what happens now...
My reasoning is was as follows:
Basically suppose there was a positive radius of convergence and call it $R>0$.
Now for $1>\lambda> 0$ set
$g_\lambda(x)=\frac{1}{\lambda} f(\lambda x)$. Clearly, $g_\lambda$ is analytic on $|x|<\frac{R}{\lambda}$. Moreover, as $\lambda\to 0$ we see that $g_\lambda\to c_1=1$ $g_\lambda(x) \to x$ (NOT $g_\lambda\to c_1=1$) uniformly on compact subsets (by Arzela-Ascoli).
Now it is clear that $g_\lambda(g_\lambda(x))=\frac{1}{\lambda} f(f(\lambda x))$
but
$\frac{1}{\lambda} f(f(\lambda x))=\frac{1}{\lambda} (\lambda x+f(\lambda x) ^2)=x+\lambda g_\lambda(x)^2$.
That is $g_\lambda$ satisfies the equation
$g_\lambda(g_\lambda(x))=x+\lambda(g_\lambda(x))^2$
Now let $\lambda\to 0$. For any $x$ with $|x|<2$ the left hand side tends to $c_1=1$$x$ (NOT $c_1=1$) while the right hand side tends to $x$. This is clearly impossible. So everything is consistent.
I think this works. Obviously not.
