There are a number of results which count the number of possible algebraic structures on a set of $n$ elements. Notable previous MO questions are for example here and here. The analogous question for finite categories does not seem to enjoy the same attention as other algebraic structures. The only source I could find was the following OEIS entry. Simply put the first question is as follows:

How many categories with at most $n$ morphisms are there up to isomorphism? Up to equivalence?

Comparing with the analogous OEIS entry for monoids, we see that for small $n$, "almost all" categories are monoids.

In general, under some appropriate weighting scheme (which?), are almost all categories monoids, in the sense that the quotient $$\frac{\text{Number of monoids with at most }n \text{ elements}}{\text{Number of categories with at most }n \text{ morphisms}}\rightarrow 1$$ as $n\rightarrow \infty$ ?

Tom Leinster asks a similar question about groups on the $n$-category Café.