`Naturally occuring'  $K(\pi, n)$ spaces, for $n \geq 2$. 
[edited!] Given a group $\pi$ and an integer $n>1$, what are examples of Eilenberg-Maclane spaces $K(\pi, n)$ that can be constructed as "known" manifolds? (or if not a manifold, say some space people had a pre-existing desire to study before $K(\pi,n)$ spaces were identified as being of interest)

Constructing $K({\bf Z}, 2)$ as ${\bf CP}^{\infty}$ is the only example I know - but there must be more out there.
I'm interested in concrete examples (like the one above) that could, e.g., be given in a Topics grad course for topology students. They seem to be scarse, so it would be nice to know what was known.
Note: I've excluded $n=1$ because most people know examples (or can figure them out) in this case.
 A: Let BTOP and BPL be the classifying spaces of topological/PL-sphere bundles and $TOP/PL$ the homotopy fiber of the map $BPL \to BTOP$. The $TOP/PL$ is a model for a $K(\mathbb{Z}/2\mathbb{Z},3)$ by Kirby and Siebenmann. This identifies a third cohomology class as obstruction to get a PL-structure on a topological sphere bundle. 
A: If $M$ is a hyperfinite type $I\!I\!I_1$ factor, then (at least conjecturally), its group of outer automorphisms is a $K(\mathbb Z,3)$.
This is based on the following three properties of that von Neumann algebra:
• The group of unitary central elements of $M$ is a circle, and thus a $K(\mathbb Z,1)$.
• The group of unitaries in $M$ is contractible.
• The automorphism group of $M$ is contractible (conjectural).
To see that $Out(M)\cong K(\mathbb Z,3)$, apply the long exact sequence of homotopy groups to the following two fiber sequences:
$$
U(Z(M)) \to U(M) \to Inn(M)
$$
$$
Inn(M) \to Aut(M) \to Out(M)
$$
As a consequence, we also get that $BOut(M)\cong K(\mathbb Z,4)$.
I recommend my talk "A K(ℤ,4) in nature"  (MSRI, April 2014), for an explanation of how to realize $Out(M)$ as the automorphism group of a naturally occurring mathemtical object.
A: Following up on Dai's answer, one can go a step further since $P U(H)$ is obviously a group.  So if we can find a contractible space on which it acts freely, the quotient will be the next level up (namely, a $K(\mathbb{Z},3)$.
Such a space can be constructed as follows: take our favourite (separable, though that's not necessary) Hilbert space, $H$, and consider $HS(H)$, the space of Hilbert-Schmidt operators on $H$.  This is isomorphic to the Hilbert tensor product $H^* \widehat{\otimes} H$ so is a Hilbert space.  Its unitary group is thus contractible.  The group $U(H)$ acts on $HS(H)$ by conjugation, and once we divide out by the centre this becomes free.  Thus $P U(H)$ acts on $U(HS(H))$ freely and so the quotient is a $K(\mathbb{Z},3)$.
However, as $P U(H)$ does not act centrally on $U(HS(H))$, the iteration stops here.
A: The following example appears in the definition of twisted $K$-theory. 
Let $H$ be an infinite dimensional separable Hilbert space over $\mathbb{C}$. Since the unitary group $U(H)$ is contractible, the projective unitary group $PU(H)= U(H)/S^1$ has the homotopy type of $K(\mathbb{Z},2)$. The fact that $BPU(H)\simeq K(\mathbb{Z},3)$ and the fact that $PU(H)$ acts on the space of Fredholm operators $\mathrm{Fred}(H)$ are essential in the definition of twisted $K$-theory.
A: There is a very nice model of $K(\mathbb Z,n)$ which is given by the free abelian topological group on the pointed space $(S^n,\star)$, let us call that $F(S^n,\star)$. An element in $F(S^n,\star)$ is given by a finite set of points in $S^n \setminus \lbrace\star\rbrace$ such that each point in this finite carries a non-zero integer as a label with the obvious addition. The topology is more subtle to describe and made in such a way that $F(S^n,\star)$ is an abelian topological group, the inclusion $S^n \subset F(S^n,\star)$ is continuous and $\star=0$ in $F(S^n,\star)$.
Though, I am not sure whether $F(S^n,\star)$ is an infinite-dimensional manifold (I think not), it is still pretty regular being a topological group and a CW-complex at the same time.
This is all very classical and was studied in detail in
Dold, Albrecht; Thom, René, Quasifaserungen und unendliche symmetrische Produkte., Ann. of Math. (2) 67 1958 239–281. 
