It goes like this: Friedman wants to establish new sovereign nations built on oil-rig-type platforms anchored in international waters—free from the regulation, laws, and moral suasion of any landlocked country. They'd be small city-states at first, although the aim is to have tens of millions of seasteading residents by 2050. Architectural plans for a prototype involve a movable, diesel-powered, 12,000-ton structure with room for 270 residents, with the idea that dozens—perhaps even hundreds—of these could be linked together. Friedman hopes to launch a flotilla of offices off the San Francisco coast next year; full-time settlement, he predicts, will follow in about seven years; and full diplomatic recognition by the United Nations, well, that'll take some lawyers and time.

"The ultimate goal," Friedman says, "is to open a frontier for experimenting with new ideas for government." This translates into the founding of ideologically oriented micro-states on the high seas, a kind of floating petri dish for implementing policies that libertarians, stymied by indifference at the voting booths, have been unable to advance: no welfare, looser building codes, no minimum wage, and few restrictions on weapons.

It's a vivid, wild-eyed dream—think Burning Man as reimagined by Ayn Rand's John Galt and steered out to sea by Captain Nemo—but Friedman and Thiel, aware of the long and tragicomic history of failed libertarian utopias, believe that entrepreneurial zeal sets this scheme apart. One potential model is something Friedman calls Appletopia: A corporation, such as Apple, "starts a country as a business. The more desirable the country, the more valuable the real estate," Friedman says. When I ask if this wouldn't amount to a shareholder dictatorship, he doesn't flinch. "The way most dictatorships work now, they're enforced on people who aren't allowed to leave." Appletopia, or any seasteading colony, would entail a more benevolent variety of dictatorship, similar to your cell-phone contract: You don't like it, you leave. Citizenship as free agency, you might say. Or as Ken Howery, one of Thiel's partners at the Founders Fund, puts it, "It's almost like there's a cartel of governments, and this is a way to force governments to compete in a free-market way."

Some experts have scoffed at the legal and logistical practicalities of seasteading. Margaret Crawford, an expert on urban planning and a professor of architecture at Berkeley, calls it "a silly idea without any urban-planning implications whatsoever." Other observers have mocked it outright, such as Slate's Jacob Weisberg, who deemed it perhaps "the most elaborate effort ever devised by a group of computer nerds to get invited to an orgy." Despite the naysayers, Thiel appears firmly committed to the idea; he has so far funneled $1.25 million to the Seasteading Institute.

"When you start a company, true freedom is at the beginning of things," he says and slides the thought over to the topic of nations. "The United States Constitution had things you could do at the beginning that you couldn't do later. So the question is, can you go back to the beginning of things? How do you start over?"

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For Thiel, ambition like this—outsize, contrarian, vaguely seditious—is nothing new. He was born in Frankfurt, Germany, but his father's career as a chemical engineer made for an itinerant childhood: He attended seven different elementary schools on two continents before the family settled in Northern California. He was a chess prodigy—at his peak he was ranked No. 7 in the U.S. Chess Federation's Under-13 bracket—and then a math prodigy at San Mateo High School. His intensity, even then, was remarkable. "He drove a 1978 Volkswagen Rabbit," recalls Norman Book, a high-school friend. "You'd always see him pulling out of the school lot, leaning way forward in the car. That's because it was a four-cylinder, and he just couldn't ever get it to go fast enough. Leaning forward like that, it was almost like he was willing it to go faster." (Years later, Thiel scratched this particular itch with purchases of a Ferrari Spyder and a $500,000 McLaren supercar.)

At Stanford, where he majored in philosophy, Thiel chafed at the identity politics that was in vogue on campus at the time—at the strictures of political correctness. "I think there's something unhealthy about anything that pushes to that much conformity," he says. He cofounded the Stanford Review, a zealously libertarian newspaper whose staff Thiel would later use as a talent pool for PayPal hiring. The Review was deliberately, even recklessly incendiary (Thiel's fondness for this approach is evident in his past funding of the guerrilla activist James O'Keefe, of ACORN sting-video fame); provocation was a primary goal. Sometimes it went too far: During Thiel's final year of law school, in what was characterized as a free-speech exercise, one of the Review's editors, Keith Rabois, shouted, "Faggot! Hope you die of aids!" outside the residence of a dorm supervisor, resulting in a firestorm that prompted Rabois to leave Stanford. Thiel, who was outed as gay in 2007, devoted several pages to the incident in The Diversity Myth, a 1995 book he coauthored, writing that "Keith did not deserve months of public condemnation and ostracism." Thiel later brought Rabois to PayPal as an executive vice president.