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 $S^n \to X$.
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/
