That is to say, can one find a good bound on $\pi_i(S^n)$? Let us assume that $i\ge 2n$ to avoid all infinite quantities. Particularly, I am interested to see if there is a bound of exponential type. I do not see a way to do this though.
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$\begingroup$ If $i$ and $n$ are both allowed to vary this is probably too hard. Potentially easier would be to ask about the stable homotopy groups $\pi_k^S = \pi_{n+k}(S^n)$, which are finite for $k > 0$, although this still might be too hard. Another interesting question would be to bound the prime divisors, or the $p$component for a given prime. $\endgroup$ – Kevin Casto Oct 2 '16 at 0:06
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There are relevant estimates in the 1986 paper by HansWerner Henn.
HansWerner Henn, MR 850372 On the growth of homotopy groups, Manuscripta Math. 56 (1986), no. 2, 235245.