Is there such thing as an $n$-fold cover of $SO(n)$? I know already that the $Spin(n)$ group is a double cover of $SO(n)$, but is there such a thing as a triple cover, fourfold or $n$-fold cover? I am interested to know if there are 3-spinors (?) or the likes of them, or is there some simple reason why such things are not possible.

No. The fundamental group of $SO(n)$ is $\Bbb Z/2$, so its universal cover is just a double cover. Indeed, what this is saying is that $Spin(n)$ is simply connected.