Take the 2-minute tour ×  Pete Clark threw down the challenge in
his comment to my answer on Why the heck are the homotopy groups
of the sphere so damn complicated?:  Have the homotopy groups of
spheres ever been applied to anything, including in algebraic
topology itself?  It started to get some answers in those
comments, but comments are a lousy place to record answers to a
question like this so I'm reposting it as a question.  In order
to add some more value to the question (and justify my reposting
it), let me say that I can foresee answers coming in several
different flavours and I'd like the answers to explicitly say
which flavour they use.  Firstly, there is the distinction
between stable and unstable homotopy groups. Briefly, there is a
natural map $\pi_k(S^n) \to \pi_{k+1}(S^{n+1})$ and eventually
(you will see the phrase, "in the stable range") this becomes an
isomorphism. Once it is an isomorphism, we refer to them as the
stable homotopy groups. So there are more unstable homotopy
groups than stabl