Behind the luxury hotels lining London's Park Lane, just across from a service entrance, Nigel Farage stood outside a squat office building streaked with soot. Britain's famous anti-European Union campaigner was flanked by a couple of minor sports celebrities and two young women in matching dresses who held up a blue ribbon. Farage lifted a pair of scissors and paused with a smile, mouth agape, his face frozen for the cameras in a silent chortle.

It was Sept. 29, 2016, three months after the U.K.'s shock vote to exit the EU, and Farage was the guest of honor at the opening of a small bookmaking shop owned by a man who calls himself "the most exclusive private bookmaker in the world." Ben Keith and his operation, Star Sports, specialize in high-end, high-profile clients. Keith, a diehard Farage supporter, was in the picture too, next to his idol.

Gambling is legal in the U.K., even corporatized, but most politicians aren't elbowing each other out of the way to help open betting shops. Farage is different. Throughout his professional and political lives, he has surrounded himself with men who make or take wagers, big and small, on everything from dog races to elections. They have played leading roles in promoting or financing Farage's ascent in the U.K. Independence Party (UKIP), the rise of his party to the center of the British political agenda, and his campaign to leave the EU, which has thrown the nation into a political crisis.