The house with two rooms, invented by Bing [1], is similar to the dunce hat: it is a topological space that is contractible in a non-obvious way, i.e. it is not collapsible (in particular, it has no boundary point from which a contraction can start).

As it is contractible, every loop in that space is contractible (the fundamental group of the space is trivial). This animation shows how to contract a loop surrounding the upper entrance of the house.

[1] Bing, R. H., Some Aspects of the Topology of 3-Manifolds Related to the Poincaré Conjecture, Lectures on Modern Mathematics, Volume 2, 1964