What are some examples of groups that can not be given a smooth structure such that they become a Lie Group?
Edit: To be a bit more specific, I was hoping that somebody could give an example of a finite dimensional topological group that is a topological manifold but does not admit a smooth structure making it into a Lie Group.
Any group is a Lie group if you give it the discrete topology. The better question is whether a topological group has a smooth structure that makes it a Lie group.
Local compactness is obviously necessary (because you want finite dimensions), so any non-locally compact group will be an example.
Generally, most locally compact groups are Lie groups. This question is essentially Hilbert's 5th Problem, which has been solved: https://en.wikipedia.org/wiki/Hilbert's_fifth_problem