On statelessness: Persons intending to renounce U.S. citizenship should be aware that, unless they already possess a foreign nationality, they may be rendered stateless and, thus, lack the protection of any government. They may also have difficulty traveling as they may not be entitled to a passport from any country. Even if they were not stateless, they would still be required to obtain a visa to travel to the United States, or show that they are eligible for admission pursuant to the terms of the Visa Waiver Pilot Program (VWPP). If found ineligible for a visa or the VWPP to come to the U.S., a renunciant, under certain circumstances, could be barred from entering the United States. Nonetheless, renunciation of U.S. citizenship may not prevent a foreign country from deporting that individual back to the United States in some non-citizen status.
http://travel.state.gov/law/citizenship/citizenship_776.html
The catch is you'll probably want to do this in a friendly country or one you're already a citizen of.
http://principality-of-sealand.eu/faq_e.html
Actually, the above website may not be from the REAL Sealand... long story.
http://www.dominica.gov.dm/cms/index.php?q=node/678
https://www.henleyglobal.com/countries/austria/citizenship/
On the other hand, dozens of countries will give you a visa in exchange for an investment; there's nothing unusual about the US in that regard, contra what forgottenpasswrd tries to imply. In some cases the "investment" is just a purchase of government bonds; in other cases it means an actual business that is capital-intensive and generates good jobs for locals. After you get the visa, you still have to actually live there for some period of time and show evidence of integration (like speaking the language) to qualify for naturalisation. The amount of money and the period of time range from four or five figures and a couple of years (Peru, Ecuador, Paraguay, Dominican Republic) all the way up to seven figures and the better part of a decade (Hong Kong, Germany).