I am wondering if there is a CW-complex $X$ of dimension $n$ with only one (dense) $n$-cell such that $H_n(X)=0$ and some homology group has non-trivial free part.
For example, I am aware that a non-orientable manifold has $H_n(X)=0$ and $H_{n-1}(X)=\frac{\mathbb{Z}}{2\mathbb{Z}}\oplus \mathbb{Z}^k$ for some $k$, but can they be achieved with only one $n$-cell if $k>0$?
Sure, the usual CW structure of the projective plane $\mathbb R\mathbb P^2$ has a single $2$-cell:
You may just wedge a $1$-cell onto this to obtain a non-trivial free part in $H_1$, since the homology of a wedge is the direct sum of the homologies.