Is there an analogue of the universal cover for higher homotopy groups? The universal cover $U$ of a topological space $X$ is a simply-connected covering space of $X$. As the 'universal' moniker implies, this space is universal in the category of covering spaces of $X$ (and thus is unique up to homeomorphism), and pleasantly, there's a free and faithful group action of $\pi_1(X)$ on $U$ with the orbit space $U/\pi_1(X) \cong X$.
Question: is there a higher-dimensional analogue of a space with some or all of these properties? Clearly a cover won't suffice, but does every space have, say, a fiber bundle over it that's $n$-connected (I feel as though this is the proper generalization rather than trivial $\pi_n$, for whatever reason) and, more importantly (to me), is universal in some sense? Do we have an analogue of the group action we had for our universal cover? If so, what other pleasant properties of the universal cover transer to this generalization?
EDIT: To be clear, this is what I want. 


*

*For each $n$, a map $U_n(X): \textbf{Top} \rightarrow \textbf{Top}$ such that...

*$U_1(X) = \tilde X$, the universal cover of $X$

*$U_n(X)=X$ if $X$ is $n$-connected.

*$U_n(X)$ is universal in some appropriate setting.

*I'm not sure fibrations are the right tool for this, but I'd like $U_n(X)$ to fiber over $X$. I'm pretty sure this gives us functoriality of $U_n$, but I didn't work out the details. 


I perhaps emphasized too much the group action; it'd be neat if an analogue existed for $U_n$, but I have no pretense that there should be.
 A: I don't know whether the following qualifies but hopefully it is helpful. (EDIT: after the OP clarified their question, it appears this answer does not qualify, but I've chosen to leave the answer here anyway in case someone gains something from it.)
Every based space $(B,\ast)$ fits into a fibration sequence $\Omega B\to PB\stackrel{p}{\to} B$ where $PB$ is the space of paths $\gamma\colon [0,1]\to B$ with $\gamma(0)=\ast$ and $\Omega B$ is the space of loops in $B$ based at $\ast$. We give these spaces the compact-open topology. The map $p$ is given by sending a path $\gamma$ to its endpoint $\gamma(1)$. The space $PB$ is contractible by simply shrinking all paths to the constant path at $\ast$.
By the induced long exact sequence in homotopy, we get the corollary $\pi_n(\Omega B)\cong\pi_{n+1}(B)$ for all $n\geq 0$.
Does the space $PB$ fit your idea for a 'universal' bundle (not to be confused with this) over $B$? Note that $p$ is not always locally trivial and so isn't a bundle in general, but is certainly a fibration.
A: This question has a nice answer on MO: this answer by Reid Barton, as mentioned by Bruno Stonek in the comments.
The Whitehead tower has most of the properties I was looking for, too; thanks to Piotr Pstrągowski for mentioning this in the comments as well. 
