Is there a ceiling on gain with feedback, even with freedom to choose $\hat{F}$?

In an idealized system you can get any gain you want by choosing $\hat{F}$ for gain $\sim \frac{1}{F}$, so long as $\vert\hat{A}\hat{F}\vert»1$. You can't get a gain larger than $\vert\hat{A}\vert$ by this method (but typically $\vert\hat{A}\vert$ is quite large, like $10^{6}$).