TITLE: Categories without a tensor product
QUESTION [6 upvotes]: What is an example of a category that is useful in geometry and which does not have a tensor product, i.e. a category which we do not know how to turn into a monoidal category? 
(I am excluding cartesian monoidal categories too.)

REPLY [12 votes]: In any monoidal category $(C, \otimes, I)$, the unit object $I$ has the property that its endomorphism monoid $\text{End}(I)$ is commutative, by the Eckmann-Hilton argument. So if $C$ is a category in which every endomorphism monoid is noncommutative, then it cannot have any monoidal structure.  
A reasonably natural geometric example is given by the category of closed connected surfaces. It's a cute exercise to show that every endomorphism monoid here is noncommutative.