In a reading course on measure theory this semester I had the pleasure of preparing a lecture covering the existence-uniqueness of Haar measure on locally compact groups. Since the proofs (as presented in Cohn's book) depend on the local compactness already at the very start, it is obvious that the reasoning and constructions presented don't work for groups that aren't locally compact.
I wonder (for the sake of curiosity, the presentation is already over and this is not homework) if there can exist Haar measures in those cases as well. Does anyone know of any examples of such groups, where existence or uniqueness doesn't hold?
I have tried searching and thinking on my own, but I have trouble coming up with groups that aren't locally compact to begin with…
If you require that a Haar measure is also a Radon measure, then yes: a topological group carries a Haar measure if and only if it is locally compact.
See 443E in Fremlin's Measure theory Vol. 4 Topological Measure Spaces.
As for examples of groups which are not locally compact, try to show that there is no Haar measure on an infinite-dimensional Banach space.