Are there any known interesting F-(co)algebras where F isn't a set endofunctor? Are there any known interesting F-(co)algebras where F isn't a $Set$ endofunctor? Every example I can think of deals with sets: an algebra of $X\mapsto 1+X$ for natural numbers, an algebra of $X\mapsto 1+A\times X$ for lists, a coalgebra of $X\mapsto A\times X$ for streams, etc. I'm not alone in this, Adámek writes

Although all important examples of application of coalgebra seem to concern
  coalgebras in Set, there are good reasons to develop the whole theory in an abstract category, e.g., [...]

in his "Introduction to coalgebra".
Surely there are interesting cases dealing with domains? Are there any dealing with the categories of topological spaces, of algebras, etc?
 A: As Mariano mentions in the comments, coalgebras over a field are in particular coalgeras over the functor $X \mapsto X \otimes X$ on vector spaces, although there is additional structure and properties here. For examples see this MO question and this MO question. 
Leinster's A general theory of self-similarity addresses the question of characterizing topological spaces as terminal coalgebras in interesting ways. Examples include $[0, 1]$, the Cantor set, and, conjecturally, all Julia sets.
If $P$ is a poset regarded as a category in the usual way and $F : P \to P$ is an endomorphism of $P$, then algebras over $F$ are elements $x \in P$ such that $F(x) \le x$, and similarly coalgebras over $F$ are elements $x \in P$ such that $x \le F(x)$. Initial algebras are then least fixed points, and terminal coalgebras are greatest fixed points; see, for example, the Knaster-Tarski theorem. 
A rich class of examples are given by coalgebras over comonads, which can be built out of any adjunction and which are therefore extremely general. 
