Throughout, let $X$ be a *connected finite* CW-complex. 



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> **Question:** If $X$ is of dimension $n$. Is there some integer $n'$ (maybe depending only on $n$), such that all homotopy groups $\pi\_k(X)$ for $k \geq n'$ are finite? 
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For the spheres $S^n$, $n'=2n+1$ works by Freudenthal's Suspension Theorem and Serre's result that the stable homotopy groups in that range are finite. More generally, if $\pi\_1(X)=0$, then the Milnor-Moore theorem relates the rational homotopy groups to the rational homology of the loop space of $X$ and I believe that this can be used to get a similar conclusion. But what if $\pi\_1(X) \neq 0$?


**EDIT:** Igor Belegradek (besides answering the question) pointed out that what I stated in the last three lines is not correct.



Finiteness of higher homotopy groups of finite complexes
The answer is no in a very strong way even for simply-connected complexes. In rational homotopy theory there is a famous dichotomy between elliptic and hyperbolic spaces: a simply-connected finite complex is either elliptic or hyperbolic. Elliptic means that all but finitely many homotopy groups are finite. Hyperbolic means that the sum of ranks of first $k$ homotopy groups grows exponentially with $k$. In some sense most spaces are hyperbolic. If I remember correctly, $m$-fold connected sum of $S^2\times S^2$ with itself is hyperbolic if $m>1$. You can read more of this in the book "Rational homotopy theory" by Felix-Halperin-Thomas.


