The previous answers gave analogues to the universal covering space and looked at the homotopy groups of these analogues.

However, the analogy to the n=1 case is not complete: While \pi_ 1(X) classifies the automorphisms over X of the universal covering space, the \pi_ n(X) don't classify the automorphisms over X of these n-connected analogues. In fact, the higher \pi_ n(X) don't seem to classify anything else than homotopy classes of maps S^n-->X.

This is one of the motivations for using n-groupoids as invariants of spaces, see the discussion here, right before the references:
http://ncatlab.org/nlab/show/fundamental+group+of+a+topos

or, starting on page 17 with a nice story on the invention of the higher homotopy groups, and the desire for a non-abelian homology:
http://www.intlpress.com/hha/v1/n1/a1/