A year ago, username @Greg Graviton asked in a thread here about the Spin group as covering of the spatial rotations.
A subquestion was: What other groups, even larger than SU(2) are there that (could) act on "physics" and are an extension of SO(3) Is it possible to classify all possibilities, in particular the ones that are not direct products?
My answer to this was: Of course, there are many more groups relevant for Quantum Mechanics and other physical theories (like gauge groups, as you mentioned). But concerning your second question, as far as "rotations only" go, SU(2) is already the universal covering group. These are the simple connected Lie-Groups, which lie on top of these upward chains you're searching for.
Now recently I came across a construction for structures lying above the spin structure, namely string stucture and then fivebrane structure. There are mainly the nLab references, and there is a thread on MathOverflow, relating the Whitehead tower to physics. Without knowing too much homotopy, I can still see the definitions seem geometrical and hence natural to me. The definitions seem to be quite general constructions over the manifold and are the discussed in the context of QFTs.
I make some observations which surprise me a little and I want to ask these specific questions:
The title of the thread tries to address this question. Their names kind of suggest it, but do these mathematical objects, the objects above the spin structure, necessarily have to do with string theory? Now this question is problematic in as far as string theory people will say QFT is generally the necessary conclusion when studying QFTs. I want to filter this thought out for a moment and ask: If you want to do generally non-stringly quantum field theory, do you for practical reasons want to spend your time learning about these objects? Do you come across them just by learning about Dirac operators?
The presentation for field theories discussing this focus on a map $\Sigma\to X$, were $\Sigma$ gives the field configuration and $X$ is a manifold. I figure I have to view this just as generalized element of the manifold. Now I find it curious that the Spin, String and fivebrane structure seems to be something build over $X$ first, prior to considering physically propagating things in that world. This reverses my thinking of the spin being something associated with the particle, not with the space it lives in. When following this approach, to what extend to I think of the spin (bundle, etc.) to be a property of the world the object lives in, and then what is the object itself, if we say it only gets these feature when it's put into $X$?
Why fivebrane? Why not some other number?