A good case to look at would be Michiel Hazewinkel's 'star example' of a rich structure **Symm**, the ring of symmetric functions in a countably infinite number of indeterminates.

> **Symm**, the Hopf algebra of the symmetric functions is a truly amazing
> and rich object. It turns up
> everywhere and carries more extra
> structure than one would believe
> possible. For instance it turns up as
> the homology of the classifying space
> **BU** and also as the cohomology of that space, illustrating its
> self-duality. It turns up as the
> direct sum of the representation
> spaces of the symmetric group and as
> the ring of rational representations
> of the infinite general linear group.
> This time it is Schur duality that is
> involved. It is the free
> $\lambda$-ring on one generator. It
> has a nondegenerate inner product
> which makes it self-dual and the
> associated orthonormal basis of the
> Schur symmetric functions is such that
> coproduct and product are positive
> with respect to these basis
> functions...**Symm** is also the
> representing ring of the functor of
> the big Witt vectors and the covariant
> bialgebra of the formal group of the
> big Witt vectors (another
> manifestation of its auto-duality)... 
> 
> As the free $\lambda$-ring on one
> generator it of course carries a
> $\lambda$-ring structure. In addition
> it carries ring endomorphisms which
> define a functorial $\lambda$-ring
> structure on the rings $W(A) =
> CRing(Symm, A)$ for all unital commutative rings $A$. A sort of
> higher $\lambda$-ring structure. Being
> self dual there are also
> co-$\lambda$-ring structures and
> higher co-$\lambda$-ring structures
> (whatever those may be).
> 
> Of course, **Symm** carries still more
> structure: it has a second
> multiplication and a second
> comultiplication (dual to each other)
> that make it a coring object in the
> category of algebras and, dually,
> (almost) a ring object in the category
> of coalgebras.
> 
> The functor represented by **Symm**,
> i.e. the big Witt vector functor, has
> a comonad structure and the associated
> coalgebras are precisely the
> $\lambda$-rings.
> 
> All this by no means exhausts the
> manifestations of and structures
> carried by **Symm**. It seems unlikely
> that there is any object in
> mathematics richer and/or more
> beautiful than this one, and many more
> uniqueness theorems are needed. ([Witt
> vectors. Part 1][1]: 7)



  [1]: http://arxiv.org/abs/0804.3888