I think one possible solution to the United States' billionaire problem and the world's billionaire problem is this proposal (not original to me):
When someone becomes a billionaire, there is a special ceremony - perhaps televised - where they are presented with a handsome trophy labeled "I won at capitalism!" Then the government takes 99.9% of their money away and gives it to some charity.
If that billion was really due to their own brains and talent, then they can do it again. Right?