I am not sure if you consider these creative but some typical examples of additive categories are
- the category $\mathcal{R}$-mod, of modules over a ring or a $k$-algebra $\mathcal{R}$,
- the category $\mathcal{Comp_\mathcal{R}}$, of chain complexes of $\mathcal{R}$-modules,
- the category $\mathcal{Ab}$ of abelian groups,
- the category of $\mathcal{H}$ of commutative, cocommutative hopf algebras, over an algebraically closed field of characteristic zero
Details on their structure can be found in most Categroy theory textbooks.
On the other hand, examples of non-additive categories are: the category of sets, the category of fields, the category of $k$-algebras, etc.