I'm looking for an example of a connected manifold $M$ (ideally a closed manifold) that requires "distinct" charts.
By requiring distinct charts I mean that there is no single coordinate chart $(U, \phi)$ on $M$ where $U$ is simply connected and dense in $M$.
I thought it would be easy to give a concrete example, but I'm struggling to come up with one. For example, the sphere $S^2$ does not work because there is the chart $(S^2 \setminus \{N \}, \phi)$ where $N$ is the north pole, and $\phi$ is the stereographic projection from the north pole.
My next idea was to consider a surface of genus $2$, but I didn't have a way of proving that such a chart didn't exist.