  We explore connected affine algebraic groups $G$, which enjoy the following
finiteness property $\rm (F)$: for every algebraic action of $G$, the closure
of every $G$-orbit contains only finitely many $G$-orbits. We obtain two main
results. First, we classify such groups. Namely, we prove that a connected
affine algebraic group $G$ enjoys property $\rm (F)$ if and only if $G$ is
either a torus or a product of a torus and a one-dimensional connected
unipotent algebraic group. Secondly, we obtain a characterization of such
groups in terms of the modality of action in the sense of V. Arnol'd. Namely,
we prove that a connected affine algebraic group $G$ enjoys property $\rm (F)$
if and only if for every irreducible algebraic variety $X$ endowed with an
algebraic action of $G$, the modality of $X$ is equal to $\dim X-\max_{x\in X}
G\cdot x$.
