  We introduce asymptotic R\'enyi entropies as a parameterized family of
invariants for random walks on groups. These invariants interpolate between
various well-studied properties of the random walk, including the growth rate
of the group, the Shannon entropy, and the spectral radius. They furthermore
offer large deviation counterparts of the Shannon-McMillan-Breiman Theorem. We
prove some basic properties of asymptotic R\'enyi entropies that apply to all
groups, and discuss their analyticity and positivity for the free group and
lamplighter groups.
