Recall that every finite-dimensional rational representation of $GL_n$ is of the form $(\det)^{-k} \varrho$ for some integer $k\geq 0$ and polynomial representation $\varrho$ (and $\det$ is the one-dimensional representation $A\mapsto \det(A)$). The irreducible polynomial representations have been classified and are given by the Schur modules.
My questions are as follows. Are there simply-described finite-dimensional non-rational representations of $GL_n$? Are there a lot of them? Can they be classified? Also, why do we care about polynomial representations in the first place?
The problem with $GL(V)$ simply as a group is that it is to big and has a lot of wild representations. For example, the field $\mathbb{C}$ has a lot of field automorphisms, using them you can twist any normal representation and obtain some strange one. So you need to add some additional structure on $GL(V)$ that you want to preserve.
Study of polynomial representations corresponds to understanding the $GL(V)$ group as an algebraic group. This is the natural choice if you want to have a theory over arbitrary (characteristic zero, algebraically closed) field. There are other versions of classification. For example, you can understand $GL_n$ as a Lie group and study holomorphic representations. The results would not change. There are a lot of results of the type "continuous representations of $U(n)$ extend to algebraic representations of $GL(n)$". So, in general, the Schur-Weil theory extends for any reasonable additional structure.
Unfortunately, I don't know the examples of ``weird'' continuous representations, but intuitevely I am confident that they do exist. For me, the representation $|det|^{\pi}$ is weird enough.