I'm looking for a natural example of a category $\mathcal{C}$ with finite limits (or just finite products) wherein some object $X$ is not isomorphic to a subobject of an inhabited object. In other words, $X$ is such that there is no $Y$ in $\mathcal{C}$ equipped with monomorphisms $1 \to Y$ and $X \to Y$. Kudos if the category is also regular.
I'm looking for a natural example of this that could occur in the mathematical wilderness. In other words, a category that a mathematician could encounter without actively looking for such a pathological example. The purpose is to give a concrete example of a regular category with finite limits that fails to have coproducts in a really pathological way.
Consider for example the category of sheaves (local homeomorphisms) over $S^1$ with connected total space.
Here the terminal object is (the identity of) $S^1$, and let $X:=\Bbb R$ with the standard (spiralic) cycling $\Bbb R\to S^1:\ \ x\mapsto e^{ix}$. If $X\subseteq Y$ is also a sheaf, and has a trivial section $S^1\to Y$, then $Y$ is not connected.