Do all manifolds have a densely defined chart? Let $M$ be a smooth connected manifold. Is it always possible to find a connected dense open subset $U$ of $M$ which is diffeomorphic to an open subset of R$^n$? 
If we don't require $U$ to be connected, the answer is yes: it is enough to construct a countable collection of disjoint open "affines" whose union is dense, and this is not terribly difficult.

Depending on what you consider a manifold, the long line may be a counterexample.
And for a non-connected manifold, surely the answer is no?  Take your favorite smooth manifold, and take a disjoint union of more than $\mathfrak{c}$ copies of it.  Again, unless your definition of "manifold" rules this out (by assuming separability, etc).
