Let us give the category of monoids $\mathbf{Mon}$ a monoidal structure with $\otimes = \sqcup$ (coproduct). How can we classify $\mathbf{CoMon}(\mathbf{Mon})$, the category of comonoids of monoids?
This category has (at least) two different descriptions, namely it is also the category of cocontinuous functors $\mathbf{Mon} \to \mathbf{Mon}$ (given by tensoring with a comonoid), but also the opposite of the category of continuous functors $\mathbf{Mon} \to \mathbf{Mon}$ (the functors represented by comonoids). From the description of cocontinous functors $\mathbf{Mon} \to \mathbf{Mon}$ we also get a structure of a cocomplete monoidal category (with $\otimes=\circ$ being cocontinous in each variable) with a zero object.
So far I have found three basic comonoids: $\langle x \rangle$ with $\nabla(x) = x_1 x_2$, $\langle x^{\pm 1} \rangle$ with $\nabla(x)=x_1 x_2$, and $\langle x \rangle$ with $\nabla(x)=x_2 x_1$. The continuous functors $\mathbf{Mon} \to \mathbf{Mon}$ represented by them are the identity functor, the group of units functor (which factors over $\mathbf{Grp}$), and the opposite monoid functor. Their left adjoints are the identity $\mathbf{1}$, the group completion $K$ and the opposite monoid functor $D$. Hence, every coproduct of them is also an example. Apart from the unit $\eta : \mathbf{1} \to K$, I think that all other morphisms between them are zero, and it seems that we have $D \circ K \cong K \circ D \cong K \circ K \cong K$.
In case it turns out that this monoidal category is too complicated, what about the category of monoids in it, i.e. the category of Tall-Wraith monoids in $\mathbf{Mon}$? Can we classify them? For $\mathbf{Grp}$ it is well-known by a result of Freyd.
