There are two extremities: on the "easy end" one has vector bundles which are classified by maps to the (more or less) well understood spaces like Grassmanians; on the "hard end" there are spherical fibrations/sphere bundles classified by maps to the classifying space of the monoid of self-homotopy equivalences of spheres.

There are ways to move "just slightly down" from the hard end, by considering self-homeomorphisms, self-PL-isomorphisms, self-diffeomorphisms, ...

On the other hand I have not seen any ways to move "just slightly up" from vector bundles. What I mean here is this: one may switch from a vector bundle to an associated sphere bundle; the corresponding sphere bundle is one whose gluing maps are linear in one sense or another ($\mathbb R$-linear, or $\mathbb C$-linear, $\mathbb H$-linear, ...). Thus there is a huge gap where on one end we have something linear and on the other something "as nonlinear as it gets".

My question is whether it is possible to squeeze in between some groups consisting of controllably nonlinear maps. Say, maps of degree $n$, with composites truncated back to degree $n$ or something similar.

Many years ago I was asking around about this, and Leonid Makar-Limanov suggested to look at maps of "finite codegree" - that is, something describable by series with vanishing low degree coefficients - since these readily form a group under composition; but this would still mean "moving down from the hard end" rather than "moving up from the easy end"...

...After some hesitation decided to add more speculative possibility that occurred when commenting to the answer below. Maybe reducing the structure group of a bundle from some linear group (say, $SU(n)$) to its increasingly highly connective covers could be interpreted as switching from linear gluing maps to ones which are "linear up to homotopy" in some sense, together with (coherent) choices of "linearizing homotopies". This brings up 2-vector bundles in the sense of Kapranov-Voevodsky or Baas-Dundas-Rognes, I wonder if this has been looked at from this angle?

Actually I am confused here by another thing, and maybe this needs separate question, but why not ask it right away? What I don't understand is this - reducing structure group to more and more highly connective covers sort of "moves towards a contractible "structure group"", while the ultimate "structure group" should be the aforementioned monoid of self-homotopy equivalences of the sphere rather than contractible. I seem to mix up different things here but how exactly?