Topological groups are homogeneous. In
J. van Mill, "A topological group having no homeomorphisms other than translations," Transactions of the AMS 280 (1983), pp. 491-498 (link),
Jan van Mill constructed an infinite topological group whose only self-homeomorphisms are group translations. Such a space is called "uniquely homogeneous" -- it is homogeneous, but for any pair of points there is exactly one self-homeomorphisms of the space witnessing homogeneity. Jan's group also has the amazing property that removing any point results in a rigid space.
In the same paper (section 4), van Mill shows that, assuming the Continuum Hypothesis, there is a topological group whose only continuous self-maps are either group translations or constant functions.
Thus the answer to your question is "consistently yes, and you can come close in ZFC." I do not know whether anyone else has come along and improved Jan's CH result to a ZFC result (but a quick glance through the papers citing Jan's seems to indicate that no one has).