Infinite "almost rigid" homogeneous $T_2$-space A topological space $(X,\tau)$ is said to be homogeneous if for all $x,y$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$.
Is there an infinite homogeneous Hausdorff space $(X,\tau)$ such that every continous map $f: X\to X$ is either a homeomorphism, or constant?
EDIT. I forgot to add "infinite" in the original question.
 A: Remark. This answer was written before the "infinite" assumption was added. 
Take the discrete space with two points. 
It is clearly homogeneous and Hausdorff, and its only self-maps are the two constant maps, the identity and the involution exchanging the two points. 
A: 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).
