Are these the only gauge-invariant functions of $A_\mu$? I know off course that $F_{\mu\nu}$ is a gauge invariant function of $A_\mu$ in the abelian case. Also we have $\epsilon^{\alpha\beta\mu\nu} F_{\alpha\beta}F_{\mu\nu}$ in that case.
Are there any other functions? Perhaps one that cannot be built up from $F_{\mu\nu}$.
In the Non-abelian case I know that $tr F_{\mu\nu}^2$ is gauge invariant. Other examples are (since $\epsilon^{\alpha\beta\mu\nu} F_{\alpha\beta}=\tilde F^{\mu\nu}$ is gauge covariant) : $tr \tilde F_{\mu\nu}^2$ $tr F_{\mu\nu}\tilde F^{\mu\nu}$.
The same question: Any other function of $A_\mu$? Perhaps one that cannot be built up from $F_{\mu\nu}$.
 A: There is a huge family of gauge-invariant operators called Wilson lines, which are of central interest to lattice QCD and holography. 
Given a path $\gamma$ in spacetime, a Wilson line $W_{\gamma}$ is a gauge-invariant operator defined as
$$W_{\gamma}\equiv\text{P}\,\exp\left\{-i\int_{\gamma}A_{\mu}\mathrm{d}x^{\mu}\right\},$$
where $\text{P}\,\exp$ is the path ordered exponential (defined in the same way as the time ordered exponential defined in basic perturbation theory). Wilson lines, unlike the operators you listed above, are non-local.
In the abelian case, note that, due to Stoke's theorem, if $\gamma$ is closed then
$$W_{\gamma}\equiv\exp\left\{-i\oint_{\gamma}A\right\}=\exp\left\{-i\int_{\Sigma}F\right\},$$
where $\Sigma$ is any surface with $\partial\Sigma=\gamma$ (note that I have used the notation of differential forms). Thus, for a Wilson line over a sufficiently tight loop, we have
$$W_{\gamma}\approx 1-iS[\gamma]F,$$
where we have approximated $F$ as roughly constant along the loop, and $S[\gamma]$ is the minimal area of a surface whose boundary is $\gamma$. Thus, for small loops, Wilson lines can produce local gauge-invariant operators.
I hope this helps!
