The Hawaiian earring, the union of a null sequence of circles joined at a common point.
Shows that a connected, locally path connected metric space can fail to be locally contractible.
(Much less obvious). Amplifies the failure of TOP as the `correct' category in which to do algebraic topology. For example the fundamental group of the Hawaiian earring (with the natural quotient topology inherited from the space of based loops) fails to be a topological group in TOP.
(The true source of pathology is not the Hawaiian earring and its properties, but rather the general failure of quotients and products to commute in the category TOP, (i.e. the quotient of the product might not be the product of the quotients with standard definitions of topological quotients and topological products). Such discrepancy makes the case for the continued relevance of category theory.