Of course it isn't really that hard - nowhere near as hard as $\pi_k(S^n)$ for $k>n$, for instance. The hardness that I'm referring to is based on the observation that apparently nobody knows how to do the calculation within the homotopy category of topological spaces. Approaches that I'm aware of include:

-Homology theory (the Hurewicz theorem)

-Degree theory

-The divergence theorem

and each of these reduces to a calculation within some other category (PL or Diff).  My question is: is there something "wrong" with hTop that precludes a computation of $\pi_n(S^n)$ within that category?

Certainly if my assumption that such a proof does not exist is wrong, I would be very interested to know it.