Question: What are examples of set-theoretic groups which do not admit any Lie group structure?
Note that this is different from asking for "a topological group which is not a Lie group", since here we would need to show that it is not a Lie group for any topology.
(Definition: I am assuming a Lie group is a second-countable Hausdorff locally-Euclidean space with a smooth structure for which the product and inversion maps are smooth.)
Note that any group with at most countably many elements can be given the structure of a $0$-dimensional Lie group with the discrete topology. Examples would thus have to be uncountable.
Take an uncountable direct sum of, say, finite cyclic groups.