$CP^\infty$ can be viewed as a CW-complex with a cell in every even dimensions. The only non-zero homotopy group is $\pi_2(CP^\infty)=Z$ which is infinite. I like to know examples (the more the better) of CW-complexes with finite homotopy groups.
For example, I like to know the number of cells in each dimensions, and the chain complex formed by those cells, so that one can compute cohomology. (I like to know the chain complex, in addition to the cohomology).