Are there any interesting examples of categories generated by a single object? By this I mean the following. Let $\mathcal{C}$ be a category and let $\star \in Ob(\mathcal{C})$. Create a new category $\mathcal{D}$ whose objects are $\star$ and formal (co)limits $F:I \to \mathcal{C}$ such that $F(i)=\star$ for all $i \in I$ (with no restriction on morphisms). We say that $\mathcal{D}$ is generated by $\star$. Do any categories we know arise in this way?
$\textbf{EDIT}:$ I think I'd prefer to drop the term "formal" from my question. I'd rather think about (co)limits that occur in $\mathcal{C}$.
For any (small) category $\mathcal{A}$, the category of "formal colimits of diagrams in $\mathcal{A}$" is equivalent to the presheaf category $\mathrm{Fun}(\mathcal{A}^\mathrm{op}, \mathbf{Set})$. (I assume you intend for the morphisms of $\mathcal{D}$ to be the ones freely generated by the condition that the objects become actual colimits in $\mathcal{D}$)
In your problem, $\mathcal{A}$ is the full subcategory of $\mathcal{C}$ spanned by the object $\star$. (I assume $\mathcal{C}$ is locally small)
In the case (such as yours) where $\mathcal{A}$ is a category with one object (which I will call $\star$), the presheaf category has an alternate description: $\hom(\star, \star)$ is a monoid $M_\mathcal{A}$, and $\mathrm{Fun}(\mathcal{A}^\mathrm{op}, \mathbf{Set})$ is equivalent to the category of right $M_{\mathcal{A}}$-sets. (i.e. sets with a right action by the monoid $M_\mathcal{A}$)
The category of "formal limits" is obtained by dualizing: $\mathrm{Fun}(\mathcal{A}, \mathbf{Set})^\mathrm{op}$.