Here is one type of example, basically inspired by the manifold-type examples but so general that they are not actually categories of manifolds. Let $B$ be any small category with finite coproducts, and let $C$ be the category of diagrams of shape

$$X \to Z \leftarrow Y$$

in $B$. Coproducts in $C$ are given pointwise by coproducts in $B$. Define

$$\partial(X \to Z \leftarrow Y) = (0 \to X + Y \leftarrow 0)$$

with the obvious extension to morphisms. It is quite clear that $\partial$ preserves coproducts and that $\partial^2 \cong 0$. Also there is a canonical natural transformation $i: \partial \to 1_C$, whose component at the object $X \to Z \leftarrow Y$ is the unique one where the arrow in the middle is the map $X + Y \to Z$ whose restrictions to $X$ and $Y$ are the given arrows $X \to Z$, $Y \to Z$ of the object.

**Edit:** As indicated in a comment below, it is simpler to consider instead the arrow category $B^{\mathbf{2}}$ as a cobordism category where $\partial(f: X \to Y) = (0 \to X)$. But a much more compelling reason to consider this construction is that, if I'm not mistaken, it satisfies a universal property as follows.

Let $\text{Coprod}$ be the 2-category of categories with finite coproducts and coproduct-preserving functors (and transformations between them); let $\text{Cobord}$ be the 2-category of cobordism categories and cobordism-preserving functors. There is an evident forgetful 2-functor

$$U: \text{Cobord} \to \text{Coprod}$$

In the other direction, the arrow-cobordism category construction defines a 2-functor

$$\text{Arr}: \text{Coprod} \to \text{Cobord}$$

and this is in fact a right 2-adjoint of the forgetful functor. (Notation: $U \dashv \text{Arr}$.) Thus $\text{Arr}(B) = B^{\mathbf{2}}$ defines the cofree cobordism category generated by a category with coproducts $B$.

In more detail, the unit $\eta: 1_{\text{Cobord}} \to \text{Arr} \circ U$ is defined componentwise as a cobordism-preserving functor $\eta C: C \to C^{\mathbf{2}}$ which at the object level takes an object $c$ to the object $i c: \partial c \to c$ in $C^{\mathbf{2}}$. (It is instructive to check the details of this.) The counit $\varepsilon: U \circ \text{Arr} \to 1_{\text{Coprod}}$ is defined componentwise as a coproduct-preserving functor $\varepsilon D: D^{\mathbf{2}} \to D$ which at the object level takes an object $g: d_1 \to d_2$ to the object $d_2$. It is reasonably straightforward to check that there are coherent isomorphisms

$$(U \stackrel{U \eta}{\to} U \circ \text{Arr} \circ U \stackrel{\varepsilon U}{\to} U) \cong 1_U$$

$$(\text{Arr} \stackrel{\eta \text{Arr}}{\to} \text{Arr} \circ U \circ \text{Arr} \stackrel{\text{Arr} \varepsilon}{\to} \text{Arr}) \cong 1_{\text{Arr}}$$

that make $\text{Arr}$ the right 2-adjoint of the forgetful functor $U$.

I think there's something deeper going on here than I understand at the present time.