Human space exploration is an ideal field for private research. There is now a body of billionaires with a geeky interest in what is out there. When you consider that any new initiative, such as a lunar base or a Mars expedition, will require assuming great personal risk, there is no Western government that would run the political risk of subjecting astronauts to a high probability of death far from Earth. Remember those long periods of space shutdown after each Shuttle accident?
Another rich field is energy research. If LFTR or thorium reactors are ever going to get built, it will be by billionaires working at offshore sites not reachable by protesters.