I already this question here but I didn't get any satisfactory answer, so I will try in MO now.

There are a lot of interesting and creative examples of categories, such as for example, the category whose objects are the positive integers and the set of morphisms from $n$ to $m$ is the set of $m \times n$ matrices with values in some ring with identity $R$.

Therefore, my question is: are there some nice (creative) examples of additive categories? In this case, could you describe their structures?