A convenient category of spectra.

In category theory, we like to work with categories satisfying some nice properties such as being Cartesian closed. If the naturally occurring category does not satifies these properties, we could try to modify it slightly to get a convenient category to work in. An example of such a category is a convenient category of topological spaces. It was proved by L.G. Lewis Jr that there is no symmetric monoidal category of spectra satisfying some natural properties that we might expect from such a category.

[This answer does not really fit the premise of the question, but still is an interesting example of a useful object that does not exist]