gr.group theory - Orders of field automorphisms of algebraic complex numbers

This amounts to the Artin-Schreier theorem, which has come up several times already on MO (c.f. Examples of algebraic closures of finite index):

if $K/F$ is a field extension with $K$ algebraically closed and $[K:F] < infty$, then
$[K:F] = 1$ or $2$, and in the latter case, $F$ is real-closed.

Thus the answer here is that $n$ can be $1$, $2$ or $infty$, and all possibilities occur: the field of real algebraic numbers gives an index $2$ subfield of $overline{mathbb{Q}}$.

(Also, just to be sure, there are elements of infinite order! E.g., if not then every element would have order $1$ or $2$, so the absolute Galois group would be abelian, and thus every finite Galois group over $mathbb{Q}$ would be abelian, and this is certainly not the case.)