I suddenly found out that in all the examples coming to my mind a (compact) smooth manifold has a "large" coordinate chart whose complement has positive codimension.
In other word, is it true that for any smooth compact manifold $M$ there exists a closed (probably singular) submanifold $N$ of positive codimension such that $M\setminus N$ is diffeomorphic to $\mathbb{R}^n$? And the same question about topological manifolds with "diffeomorphic" replaced by "homeomorphic" and the complement of positive codimension replaced by some appropriate notion of "small" subset.
Yes. You can take for $N$ the cut-locus of a point. It is of zero Lebesgue measure.