Categorization and persistent identity are two common examples:

Whenever one tries to define a simple concept, such as chair, for instance, one can always find an instantiation of said category which contradicts with any definition that may be offered. Philosophy has so far been unable to convincingly resolve the problematic of categorization.

Similarly, the question of persistent identity - how I know that I am the same person that I was a second ago, or ten years ago - poses an equally difficult challenge.