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). This is equivalent to the localization of $\mathcal{Comp(\mathcal{A})}$ with respect to the chain homotopy equivalences,
- the derived category $\mathcal{D(A)}$ of $\mathcal{A}$. This is equivalent to localizing $\mathcal{Comp(\mathcal{A})}$ with respect to quasi-isomorphisms of $\mathcal{Comp(\mathcal{A})}$ (or to localizing $\mathcal{K(A)}$ wrt to quasi-isomorphisms in $\mathcal{K(A)}$),
- the category $\mathcal{Ab}$ of abelian groups,
- the category $\mathcal{H}$ 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 (and the linked answer above).
On the other hand, examples of non-additive categories are: the category of sets, the category of fields, the category of $k$-algebras, etc.