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 $\mathcal{Comp(\mathcal{R}-mod)}$, of chain complexes of $\mathcal{R}$-modules,
- the category $\mathcal{Comp(\mathcal{A})}$, of chain complexes in an additive category $\mathcal{A}$,
- the localization $\mathcal{S}^{-1}\mathcal{A}$, where $\mathcal{A}$ is an additive category and $\mathcal{S}$ is a localizing class of morphisms,
- the homotopy category $\mathcal{K(A)}$ (with $\mathcal{A}$ an additive category),
- the derived category $\mathcal{D(A)}$ of $\mathcal{A}$,
- 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 Category 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.