There is a short argument using Zorn's lemma and the compactness of $[0,1]$, that shows every manifold must have maximal open simply connected subspaces.
However, I am wondering if it is necessarily the case that these subspaces are dense. It seems quite obvious to me that they must be dense, but am having some difficulty with the proof. Is this true, or do I lack the imagination to come up with a counterexample?
As Nate pointed out, I should have required the manifold to be connected.
The following answers a weaker form of your question, namely, is it true that every connected topological manifold contains an open dense simply-connected subset? The answer to this question is positive and even more is true:
Theorem. Every connected topological $n$-manifold contains an open dense subset homeomorphic to $R^n$.
This follows from a theorem by R.Berlanga "A mapping theorem for topological sigma-compact manifolds", Compositio Math, 1987, vol. 63, 209-216.
Berlanga's theorem generalized an earlier work by M.Brown, who proved the same theorem for compact topological manifolds. (The case of triangulated manifolds is easy but serves as a guideline for proofs in the topological category.)
Berlanga's theorem does not answer, however, the question if every maximal simply-connected open subset is dense in $M$.