Are there (preferably non-pathological) examples of smooth manifolds, which are groups, but not Lie groups?
In books one can see plenty of examples of Lie groups, but I haven't seen an example where a group is a manifold, yet its group actions are discontinuous somewhere. I have seen similar questions for topological groups, but they don't have to be locally Euclidean, so the situation is quite different.
Positive-dimensional smooth manifolds have cardinality $\Bbb R$. Pick a bijection $f: M \to \Bbb R$ and define the group operation by $g\cdot h = f^{-1}(f(g)+f(h))$. You may verify that this defines a group. In general, you can pull back any structure you like along a bijection.
You don't see people talking about it because it's not a very natural question. If you're not even preserving the topological structure, then your question actually has nothing to do with smooth manifolds. You're just asking "What are some groups of the same cardinality as $\Bbb R$?"
If you want a topological group structure on a manifold, then that's automatically a Lie group structure (for some smooth structure on the manifold).
