Riemann surfaces provide interesting examples of 1-types - interesting as they have roles in diverse areas. However, apart from 2-dimensional lens spaces, I can't readily bring to mind natural examples of spaces with non-trivial first two homotopy groups (non-trivial firt $k$-invariant optional, I suppose). Given a crossed module one can get an interesting 2-type, but this is via geometric realisation, so hardly finite-dimensional and not smooth in the usual sense (maybe in some exotic notion of smoothness).

Do natural examples of spaces with interesting 2-types turn up anywhere?

I tried to find out if I could construct one in a naive way at this question, but it fell over.