If that incredibly wealthy person goes and spends his money ostentatiously -- basically why we generally hate the rich in the first place because of the social stratification caused by them have bigger and better things -- then a sales tax with luxury tax or high profile items would quickly bring them back into the category of those most taxed.
If that incredibly wealth person didn't spend a lot of money and lived well below his means, then why do we care? He is basically producing for free for the rest of use to consume. If he invests his money or just puts it in a bank account, he is helping to increase capital formation and drive society forward. Even if he just burns his money or buries it, he is basically giving everybody else in the world more purchasing power -- and in this case he literally is working for free. I never understood this who argument about the rich sitting on their money: it means they are working without consuming and that means they really aren't being paid.
Income taxes are okay to an extent too. But they are just so much more difficult to get right. Defining income is difficult, and some people have very volatile income streams earning $200k one year and $50k the next. Should they really be taxes more (and a great deal more) than the guy who make $125 in two years? A sales tax or VAT is just easier from an implementation point of view. And there is nothing stopping you from excluding items like food and medical bills and nothing stopping luxury taxes either.