[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 $K(\mathbf Z, 2)$ as $\mathbf{CP}^{\infty}$ 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.