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. 



Is there a bound on the growth of homotopy groups of spheres?
There are relevant estimates [in the 1986 paper by Hans-Werner Henn.](https://www.evernote.com/shard/s24/sh/f3bd9a0b-63d3-41ad-bf28-1473a8ee4cfd/a913c5482ab1fd26379e369f60403e59)


*Hans-Werner Henn*, MR 850372 [**On the growth of homotopy groups**](http://dx.doi.org/10.1007/BF01172158), *Manuscripta Math.* **56** (1986), no. 2, 235--245.


