The algebraic (multiplicative) group $G^m$ acts on $\mathbb{A}^n$ (diagonally) and the quotient of $\mathbb{A}^n\setminus \{0\}$ by $G_m$ is $\mathbb{P}^{n-1}$ (which is a proper variety). I would like to find a "more interesting" example of an action of an algebraic groups $G$ on $\mathbb{A}^n$ such that the quotient of $\mathbb{A}^n\setminus A$ by $G$ is proper, where $A$ is a certain "small" subvariety of $\mathbb{A}^n$ (say, over the field of complex numbers). Under these conditions does the quotient have to be toric (and does $G$ have to be a torus)?

I would be deeply grateful for any hints or references (and I know very little on algebraic groups and toric varieties)!