It seems like there are two basic sources of examples:

1.

The basic structure you have on a space (set, scheme...) is the diagonal morphism $\Delta :X \to X\times X$. Functions on spaces are contravariant, which is why functions on a space form an algebra: $f.g = \Delta ^\ast (f \boxtimes g)$. We also have the morphism $\pi : X \to pt$, and the unit in this algebra is $\pi ^\ast 1$.

If we chose some covariant linearization of our space (like measures, topological chains...) then this space would be a coalgebra, with comultiplication given by $\Delta_\ast$, and counit $\pi_\ast$. So, for example we have the coalgebra $C_\ast (X)$ of (say, singular) chains on a topological space.

Such coalgebras are naturally cocommutative.

In nice cases, we have a pushforward and a pullback (e.g. functions on a finite set, equipped with a measure), and the algebra and coalgebra structures together form a Frobenius algebra (the inner product is $\pi _\ast \Delta ^\ast$.

2.

If our space $X$ is a group (or maybe just a monoid), then we have a multiplication map $m: X\times X \to X$, and then $m^\ast$ equips the space of functions (or, e.g. cochains) with the structure of a colagebra. If the multiplication on $X$ has a unit $e: pt \to X$, then $e^\ast$ is the counit of this algebra.

This coalgebra will not be cocommutative in general (unless $(X,m)$ is commutative).

If we also remember the ordinary multiplication structure (coming from $\Delta ^\ast$), then these two structures toegether form a Hopf algebra (also using the inversion $i:X \to X$ in the group).

Julian's example is of this second kind.

All the examples I know, are morally either of type 1 or type 2, but sometimes you need to have a very broad definition of "function"or "measure"!