TITLE: What are the ramifications of the fact that the first homotopy group can be non-commutative, whilst the higher homotopy groups can't be?
QUESTION [18 upvotes]: Does this mean that the first homotopy group in some sense contains more information than the higher homotopy groups? Is there another generalization of the fundamental group that can give rise to non-commutative groups in such a way that these groups contain more information than the higher homotopy groups?

REPLY [4 votes]: These problems puzzled the early topologists: in fact Cech's paper on higher homotopy groups was rejected for the 1932 Int. Cong. Math. at Zurich by Hopf and Alexandroff, who quickly proved they were abelian. We now know this is because group objects in groups are abelian groups. However group objects in the category  of groupoids are NOT just abelian groups, but are equivalent to crossed modules, which occurred in the 1940s in relation to second relative homotopy groups, $\pi_2(X,A,x)$. It turns out that there is a nice double groupoid  $\rho_2(X,A,x)$ consisting of homotopy classes of maps of a square $I^2$ to $X$ which map the edges to $A$ and the vertices to $x$. (The proof that the compositions are well defined is not quite trivial!). Using this Philip Higgins and I proved a 2-d van Kampen Theorem, published in  Proc. LMS 1978, i.e. 34 years ago, from which one can deduce new results on the nonabelian second relative homotopy groups, as crossed modules over the fundamental group. 
This is the start of using strict higher homotopy groupoids for obtaining  nonabelian calculations in higher homotopy theory -- see the web page in my comment, and references there. 
This idea came from examining in 1965 a proof of the 1-dim van Kampen theorem for the fundamental groupoid, and observing that it ought to generalise to higher dimensions if one had the right homotopical gadgets.  It took years to get the idea that this could be done for pairs of spaces, filtered spaces, or $n$-cubes of spaces, but apparently not easily just for spaces, or spaces with base point.