Categories without a tensor product 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.)
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.
Would you be interested categories with too many tensor products? (I'd argue that having too many choices can sometimes be similar to having no choices at all, depending on your purpose.)

Our varieties are smooth. The derived category of a coherent sheaves of variety has a tensor product, but monoidally derived equivalent varieties are isomorphic, by Balmer reconstruction. So if you want to consider varieties up to derived equivalence (and have this be a weakening of isomorphism) then you want to forget the natural monoidal structure. So, this category has a tensor product, in fact many tensor products, but frequently we don't want them. 

Here is the paper: https://arxiv.org/abs/math/0111049

(In the smooth case, all complexes are perfect, so the bounded derived category of perfect complexes is the same as the usual bounded derived category.)

I'm not expert on this though (I'm not an expert on anything, but this stuff especially so), so make up your own mind.