Karl Schroeder wrote a great blog piece on what he would do with a billion dollars. Of course with a billion dollars you can do a lot of pointless things like smash helicopters together for personal entertainment or buy ad space during the Superbowl to just stare at the camera (if you watch the Superbowl you will know if I ever get stupidly rich!) but you could also do some really useful things.
Schroeder talks about five areas of research he would like to see get 200 million dollars of research money to try to get breakthroughs: Better governance, CO2 capture, Microwave space launching, nuclear fusion and vertical farm technology. These are all fine things to research but I wouldn't focus on these in particular. I honestly think that maximizing space launch efficiency is not a useful thing to be spending extra money on at the moment. There are things we could learn from it certainly but I don't especially think it is worth the cost. I would definitely throw thorium power up there on the list as I suspect it would be a reality with that amount of money thrown at it and it might well be a magic wand that lets us prevent climate change, improve air quality and get good, clean, safe power to the entire world in a single stroke.
I wonder what the relative benefits of spending that kind of money on things like research would be vs. charity. I could feed a lot of people with a billion dollars but I could also make the world a much better place in the long term by improving systems of government or making thorium power a reality. I suspect that comparing research to concrete assistance in any numeric fashion is an exercise in futility though since first you would have to figure out just what the research will produce which seems somewhat problematic to say the least.
It makes me shake my head when I see the size of various people's fortunes. The local supermarket had a rag talking about Will Smith getting divorced and the fight over his $520 million fortune. Whether or not any of the gossip is true it seems like that the net worth guess is relatively on target and that just blows me away. Everybody in North America gave Will Smith a dollar and now he has half a billion dollars. He could personally fulfill 2 of Schroeder's wish lists and still have an unfathomable fortune remaining. Do it Will Smith! Make those scientists research something! Anything!