Naked singularity
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In general relativity, a naked singularity is a gravitational singularity without an event horizon. The singularities inside black holes are always surrounded by an area which does not allow light to escape, and therefore cannot be directly observed. A naked singularity, by contrast, is a point in space where the density is infinite and which is observable from the outside.
The theoretical existence of naked singularities is important because their existence would mean that it would be possible to observe the collapse of an object to infinite density.
Computer simulations of the collapse of a disk of dust have indicated that these objects can exist, and thus the Cosmic censorship principle (stating that singularities are always hidden) does not hold. Stephen Hawking lost a bet about this question.
This is, of course, an example of a mathematical difficulty (divergence to infinity of the density) which reveals a more profound problem in our understanding of the relevant physics involved in the process. A workable theory of quantum gravity would be very useful for solving problems such as this.