Does every group have a finite index subgroup (besides the whole group)? I suspect the answer is no, but I haven't found an example.
What about finitely presented or generated groups? Or groups with torsion?
I tried something like taking a presentation and only selecting some of the generators, but I don't this works even if $G$ has torsion, since the other generators could interact in a complicated way. Perhaps if we could find a minimal presentation of a group with torsion containing (as a generator) a torsion element, then this would work, but I'm not sure why such a minimal presentation should exist.