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: 7)