I have heard it stated vaguely more than once in discussions that before the work of Serre on homotopy groups of spheres, the answer to, for given natural numbers $n$ and $k$, the question "What is $\pi_{n+1+k}(S^n)$?" might "in principle have involved an infinite amount of data", and Serre's theorems showed that it does not.
Questions.
(0) Is there any sense in such "some of the answers might have involved infinitely much data" statements?
(1) Has anyone anywhere written something precise about this, i.e. about what mathematicians before Serre feared could have been the sort-of-worst state of affais about the, already-then-known-to-be-abelian, homotopy groups of spheres?
Remarks.
Needless to say, it seems a fact that for about two decades before Serre it was known that, for more general reasons, all homotopy groups of spheres are abelian.
Saying "The abelian group $\pi_{n+1+k}(S^n)$.", to me, is an "answer" of sorts, in a sense already comprising finitely-much data only, so it seems there is not much sense in the above-mentioned statements.
Most of the modern literature on computability of homotopy groups does not shed much light on this question, already because of the fact that since the work of Serre it is known that the answer is "no": frighteningly difficult and forever challenging as the calculations may be, the answer comprises a finite amount of data, since the homotopy groups of spheres are all finitely-generated abelian groups (and most of them even finite).
So the question in deciding whether the talk about the "infinite amount of data" is meaningless seems to involve to ascertain how complicated an abelian group a homotopy group of spheres could still have been before Serre.