Is there an object/idea in Mathematics that isn’t classified?

Meaning, for example one may be studying vector spaces and wants every two basis vectors to be normal to each other. That can be expressed as following:
$$\langle e_i , e_j \rangle = \delta _{ij}$$

And later on one can classify $\delta _{ij}$ to be a billiniar map from the vector space $V$ to $Hom(V,\mathbb{F})$ or a rank two tensor. (Where $\mathbb{F}$ is the underlying field)

Then this object has/obeys some extra/more structure, which generalizes it. Same can go for a lot of objects: transformations and matrices expressed as groups, surfaces curves as manifolds/topological spaces and so on..

So is there an object that in a sense “exists on its own”, that is: isn’t classified?

(Pic unrelated)