On the afternoon of April 27, 2006, Jeb Corliss rode the elevator up to the 86th floor of the Empire State Building wearing a fat suit and a stick-on handlebar mustache.

This was nothing special for the 34-year-old Corliss, the world’s most famous BASE jumper. Over his 14-year career, he has jumped off some of the world’s most iconic structures, including the Petronas Towers in Malaysia, the Golden Gate Bridge, and the Stratosphere casino in Las Vegas (smuggling his parachute through security inside an enormous stuffed puppy). He has jumped down the inside of the Eiffel Tower — turning a couple of twisting reverses on the way just to make it more exciting. Compared with those, the Empire State Building leap would be cake. A parachute was hidden inside the fat suit, but as he scrambled over the safety fence on the observation deck and prepared to jump, two security guards grabbed him through the rails. He was arrested and charged with “reckless endangerment with depraved indifference to life.” The last part of the charge particularly irked him: He had made sure to time his jump so he’d land on Fifth Avenue when the lights were red. In the ensuing media frenzy, an unrepentant Corliss appeared everywhere from the front page of the New York Times to Geraldo; later, he chatted with the Today show while flying in a wingsuit above Florida.

“My life is very, very simple,” he likes to say. “Life is a bunch of experiences you have…until you die. And my goal is to make those experiences as amazing as I possibly can.” He fully believes, he says, that “everybody has a gift, something they’re good at — and my gift is fear. I can do things with fear. When most people are crippled by fear, on the ground, puking, that’s when I’m at my best.”

The Empire State debacle turned out to be bad news for the small, secretive world of BASE jumping — the New York City Council passed a law banning the sport in all five boroughs. But the attention was...