How many categories have only two objects: I'm thinking that the answer is three:
Am I right or I can have an infinite number, maybe putting arrows as many as I want?
On
Remember that the source and target of an arrow do not determine the arrow! For example, consider the category with a single object $*$ and a morphism $z:*\rightarrow*$ for each $z\in\mathbb{Z}$ with composition given by $$m\circ n=m+n.$$ (As an aside, note that this looks a lot like the group $(\mathbb{Z}, +)$; indeed, every group can be thought of as a one-object category in this way!)
There are infinitely many categories with one object. They correspond to monoids. Infinitely many? Even up to isomorphism, they form a proper class!
The same is true for two-object categories.