Assume that $f(x)$ and $g(x)$ are analytic and that $f$ has a fixed point, without loss of generality $f(0)=0$ and therefore $f(f(0))=g(0)=0$. The Taylors series of $g^n(x)$ can be constructed. using Bell polynomials of iterated functions and then the iterate can be evaluated at $1/2$.
Let $$f(x)=g^{1/2}(x)$$ then \begin{eqnarray} Dg^n(0)&=&g'(0)^n \\ &=&g_1^n = g_1^{1/2} \end{eqnarray} For the second derivative \begin{eqnarray} D^2g^n(0)&=&g_2g_1^{2n-2}+g_1 D^2g^{n-1}(0) \\ &=&g_1^0g_2 g_1^{2n-2} \\ &&+g_1^1g_2 g_1^{2n-4} \\ &&+\cdots \\ &&+g_1^{n-2}g_2 g_1^2 \\ &&+g_1^{n-1}g_2 g_1^0 \\ &=&g_2\sum_{k_1=0}^{n-1}g_1^{2n-k_1-2} \end{eqnarray}
See Bell polynomials of iterated functions and What are efficient ways to compute the derivatives of iterated functions? for more details.