Analogue to covering space for higher homotopy groups? The connection between the fundamental group and covering spaces is quite fundamental.  Is there any analogue for higher homotopy groups?  It doesn't make sense to me that one could make a branched cover over a set of codimension 3, since I guess, my intuition is all about 1-D loops, and not spheres.
 A: Rather than just taking homotopy groups for a single dimension, you can also think about the kind of algebraic entity that detects homotopy in two consecutive dimensions, or indeed any number of consecutive dimensions. In this discussion, following on from an earlier post, we're looking at the fundamental 2-group which picks up homotopy in dimensions 1 and 2.
A: There's certainly a homotopy-theoretic analogue. A universal cover of a connected space $X$ is (up to homotopy) a simply connected space $X'$ and a map $X' \to X$ which is an isomorphism on $\pi_n$ for $n \geq 2$. We could next ask for a $2$-connected cover $X''$ of $X'$: a space $X''$ with $\pi_kX = 0$ for $k \leq 2$ and a map $X'' \to X'$ which is an isomorphism on $\pi_n$ for $n \geq 3$. The homotopy fiber of such a map will have a single nonzero homotopy group, in dimension $1$ - it will be a $K(\pi_2X, 1)$. (For the universal cover the fiber was the discrete space $\pi_1X = K(\pi_1X, 0)$.)
An example is the Hopf fibration $K(\mathbb{Z}, 1) = S^1 \to S^3 \to S^2$.
Geometrically it's harder to see what's going on with the $2$-connected cover than with the universal cover, because fibrations with fiber of the form $K(G, 1)$ are harder to describe than fibrations with discrete fibers (covering spaces).
A: In my 2010 PhD thesis I carried out a little bit of this program in a pedestrian way for $n=2$, and constructed a functor from suitably locally connected topological spaces to topological groupoids, which returned the universal 2-connected cover. Philosophically the construction is the same as the idea with animated technology as outlined in Peter Scholze's answer, but a bit more rigid, as we know the corresponding stack on spaces is a topological stack, and the functoriality is with functors between topological groupoids, not just between topological stacks.
The objects corresponding to covering spaces that I consider are, more or less, topological stacks over the base space where the fibres are topological stacks presented by (topologically) discrete groupoid: there is a cover of the stack by a discrete space such that the fibre product of that with itself is again a discrete space. There is also a local triviality condition. Though I didn't use this language explicitly, but the equivalent language of working with topological groupoids and anafunctors throughout.
For example, for a simply-connected topological space $X$, the universal 2-covering space is a $\pi_2(X)$-gerbe on $X$.
Added The universal 2-covering space (in my sense) of a pointed, path-connected space $(X,x)$ is a principal $\Pi_2(X,x)$-2-bundle, where $\Pi_2(X,x)$ is the fundamental groupoid of the loop space of $X$, equipped with the monoidal structure arising from loop concatenation (or equivalently the automorphism 2-group of $x$ in the fundamental bigroupoid of $X$). Here $\Pi_2(X,x)$ is given the discrete topological structure.
And the construction works in the smooth category as well, though I haven't published this, giving a Fréchet–Lie groupoid over a manifold $M$ that is a locally trivial bundle for $\Pi_2(M,x)$ (assuming $M$ is connected). This is the real point here, since there are already functorial constructions in the topological (and, now, condensed) world. This is not just a stack on the site of manifolds, or something diffeological, but a canonical presentation that is strictly functorial.
A: My apologies for coming back to this old question, but I want to address a point that I think is not really addressed so far. Namely, for $n=1$, the universal cover of a (reasonable) topological space $X$ is still a well-defined topological space $\tilde{X}$ over $X$. On the other hand, for $n>1$, the $n$-connected cover $X_n$ of $X$ is only defined as a topological space up to homotopy. But I think much of the appeal of universal covers comes from them being actual topological spaces -- certainly it feels somewhat awkward to say that the inclusion of the base point $\ast \to S^1$ is the universal cover of $S^1$, although this is certainly true in the homotopy category (as $\mathbb R\to S^1$ is the universal cover, and $\mathbb R\cong \ast$ in the homotopy category). To add to the (my?) confusion, many of the constructions here are (wonderful!) explicit constructions of such $X_n$ as topological spaces, leaving the issue of well-definedness up to homotopy somewhat implicit.
So the answers above are really interpreting the question to take place in the world of homotopy types (also variously called spaces, $\infty$-groupoids, or anima). For any such (connected, pointed) $X$, one can indeed form $\tau_{\geq n+1} X$ canonically -- in the language of $\infty$-groupoids, one is simply "discarding all low morphisms".
With this answer, I would like to advertise a slightly different point of view in which it is possible to combine topological information with homotopical information, but treating them as separate directions, and thereby actually getting an answer to the question for general $n$ that does reduce to the known answer for $n=1$.
In the world of condensed mathematics of Clausen and myself (and the closely related pyknotic mathematics of Barwick and Haine) topological spaces are replaced by condensed sets, which are sheaves of sets on the site of profinite sets (with finite families of jointly surjective maps as covers). For the kind of topological spaces discussed here, the translation is extremely mild, and CW complexes embed naturally into condensed sets just as they embed into topological spaces. On the other hand, it is natural to generalize condensed sets into condensed homotopy types, i.e. (hypercomplete) sheaves of homotopy types on profinite sets, in the sense of higher topos theory. Then just like sets embed into condensed sets as the "discrete" objects (=constant sheaves), homotopy types embed into condensed homotopy types as the "discrete" objects. Note that this clashes with terminology often used for homotopy types (and inspired by their relation to topological spaces), where the discrete homotopy types are the $0$-truncated ones, i.e. sets. I'll use the word $0$-truncated instead, so $0$-truncated condensed homotopy types are condensed sets.
In summary, condensed homotopy types contain both condensed sets(~topological spaces) and homotopy types fully faithfully, as the $0$-truncated, resp. discrete, objects.
For a CW complex $X$, there are two ways to associate to it a condensed homotopy type. On the one hand, $X$ is a (non-discrete) condensed set, and hence a condensed homotopy type that is $0$-truncated. On the other hand, $X$ defines a homotopy type $|X|$, and thus a (discrete) condensed homotopy type. One can actually describe the functor $X\mapsto |X|$ directly in this language.

Proposition. For any CW complex $X$, there is a universal homotopy type $|X|$ equipped with a map $X\to |X|$ in condensed homotopy types. This $|X|$ is the usual homotopy type associated to $X$.

(For a proof, see Lemma 11.9 here.)
For example, if $X=S^1$ is a circle, then $|X|$ is the homotopy type $K(\mathbb Z,1)$ of $S^1$, which is a point with an internal automorphism. The map $X\to |X|=K(\mathbb Z,1)$ then classifies a $\mathbb Z$-torsor $\tilde{X}\to X$ above $X$. Well, this $\tilde{X}\to X$ is actually the universal cover $\mathbb R\to S^1$!
More generally, for any connected CW complex $X$, picking a point $\ast \to |X|$, one can form the pullback
$$\tilde{X} = X\times_{|X|} \tau_{\geq 2} |X|$$
of the $1$-connected cover $\tau_{\geq 2} |X|\to |X|$ in the world of homotopy types, to the world of condensed homotopy types. In this case (as the fibres of $\tau_{\geq 2} |X|\to |X|$ are $0$-truncated), this pullback is actually a condensed set $\tilde{X}$, and this recovers the usual topological space.
Now, for general $n$ one can form the pullback
$$X_n=X\times_{|X|} \tau_{\geq n+1} |X|$$
to get some completely canonical condensed homotopy type $X_n\to X$. Note that $X_n$ will in general be neither $0$-truncated nor discrete, so it's something that nontrivially combines topological and homotopical information.
One can even go all the way to $n=\infty$ and consider
$$X_\infty = X\times_{|X|} \ast.$$
Then $X_\infty$ consists of (actual) points $x\in X$ together with a (homotopical) path connecting $x$ to $\ast$ in $|X|$.
As a word of warning, I should however say that the nice explicit constructions above are not directly tied to these $X_n$'s (they are actual topological spaces/condensed sets, after all); but all of them map canonically to $X_n$. So in this sense $X_n$ is actually a canonical recipient for all of them! It just doesn't exist in the usual world.
A: Just like there is a universal cover of every space, there is a natural $n$-connected space $X_n$ that maps to any space $X$.  To construct this space, you can add cells of dimension $n+2$ and higher to $X$ to get a space $Y$ together with a map $X \to Y$ which is an isomorphism on $\pi_i$ for $i \leq n$, but such that $\pi_i(Y)=0$ for $i>n$.  The homotopy fiber $X_n \to X$ of this map is then the "$n$-connected cover" of $X$; $X_n$ is $n$-connected but has the same homotopy groups as $X$ above $n$, as can easily be seen from the long exact sequence of the fibration. Details of this, as well as a proof of uniqueness of the $n$-connected cover, are in Hatcher starting on page 410.
More generally, if you started with an $(n-1)$-connected space, you could both kill the homotopy groups of $X$ above $n$ and kill a subgroup of $\pi_n(X)$, and then the homotopy fiber would be an "$n$-cover" of $X$ corresponding to that subgroup of $\pi_n(X)$.
A: Since Peter Scholze in his answer used his condensed framework to consider higher covering spaces treating the topological and homotopic information separately, perhaps an alternative approach to doing this should also be mentioned, alluded to in comments there, namely, cohesive homotopy types. The constructions are very similar, see sections '5.2.5 Universal coverings and geometric Whitehead towers' and '6.3.12 Universal coverings and geometric Whitehead towers' of Urs Schreiber's 'Differential cohomology in a cohesive ∞-topos'.
A: 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/
A: There may be a geometric but partial answer to your question. This is an idea I learnt from Dennis Sullivan. As we know, passing from $X$ to its universal cover $\widetilde{X}$ kills the fundamental group. Now, by Hurewicz we can assume that $H_2(\widetilde{X})=\pi_2(\widetilde{X})$, whence killing $H_2(\widetilde{X})$ suffices. If we assume $H_2(\widetilde{X})$ is torsion free then each generator $\alpha_i\in H_2(\widetilde{X})$ corresponds to a circle bundle $E_i$ over $\widetilde{X}$, i.e., $H_2(E_i)=H_2(\widetilde{X})/\mathbb{Z}\alpha_i$. Thus, if $\widetilde{X}$ was a manifold of dimension $n$ and $H_2(\widetilde{X})$ was free of rank $k$ then taking successive circle bundles we get a manifold $E$ of dimension $n+k$. This has the same higher homotopy groups ($\pi_i$ for $i>2$) as that of $X$. The example given by Reid Barton is an illustration of this. However, for manifolds this is as far as you can go since killing even the free part of $\pi_3(\widetilde{X})$ (or, equivalently the free part of $H_3(E)$) requires bundles over $E$ with fibre $\mathbb{CP}^\infty$, which lands us outside the realm of finite dimensional manifolds. 
