  There is a countable metrizable group acting continuously on the space of
rationals in such a way that the only equivariant compactification of the space
is a singleton. This is obtained by a recursive application of a construction
due to Megrelishvili, which is a metric fan equipped with a certain group of
homeomorphisms. The question of existence of a topological transformation group
with the property in the title was asked by Yu.M. Smirnov in the 1980s.
