Analogue to covering space for higher homotopy groups?

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).

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 $\pi_n(X)$.

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.