The only thing he's not losing ZZZs over? His longtime relationship with former tennis star Anna Kournikova, 26. "Anna's really cool—she's a great girl," he says. "She's the only serious girlfriend I've ever had in my life. We talk all the time. And we see each other a lot. If I'm away from home for two weeks, I always come back, even if it's for a day." And while Iglesias recently generated headlines when he told a Swedish newspaper that he and Kournikova had divorced, there is no cause for concern. In fact, they were never married in the first place—despite previous reports to the contrary. "I was [joking] around," says Iglesias. "You get asked all these personal questions, and a lot of times I've just got to put some humor in it to deal with it. How many times can I say I'm not married?" Kournikova agrees: "He's got a really dry sense of humor and sometimes people don't really get it," she says. "He's a joker. He's too much fun. That's why I like it."
Perhaps the point of confusion is the six-carat pink diamond ring Iglesias gave Kournikova a couple of years ago, but he scoffs at the idea. "So what? Girls like diamonds. A diamond ring doesn't mean you're married. I promise I never looked at it that way when I gave it to her." So will they wed anytime soon? Says Iglesias: "I'm cool for now."
Besides, the singer has been otherwise engaged: He spent the past three years working on Insomniac. "This album gave me anxiety," says Iglesias. "I didn't know what to write about anymore. The writer's block drove me nuts." Occasional breaks to waterski and surf with pals in Cabo San Lucas, Mexico, helped clear his mind. He also tackled another source of stress: his fear of flying. "I get freaked out by the lack of control and the lack of information," he explains. Taking matters into his own hands, Iglesias has learned to fly his own plane, an ultralight two-seater craft. While cruising between his home in Miami and Key Largo, Fla., he sometimes listens to Pink Floyd and Seal on his iPod, though he admits, "I know you're not supposed to use an iPod when flying. You should be more aware of what's going on." And, like his famous father, singing legend Julio Iglesias, 63, Enrique now owns (but doesn't pilot) a Gulfstream IV jet.
Yet father and son don't appear to have much more in common. Enrique says he last saw his father when his grandfather died in Spain in 2005. And he hasn't met his newest half-brother Guillermo, born to Julio and his girlfriend of 17 years, Miranda Rijnsburger, 42, in Miami Beach May 5. (Enrique has two older siblings—their mother is Julio's ex Isabel Preysler—and seven half-siblings.) "It's probably my fault because I left home at 19 and in a way, I never turned back," says Enrique. "My dad is pretty much of a control freak. And I feel like I was probably the first thing in his life he couldn't control." Still, Enrique adds, "For a lot of people, it's difficult to understand. But my dad and I really get along. When we see each other we laugh." Says Julio: "We speak sometimes—it's hard because he is traveling all the time."
During his interview with PEOPLE, Iglesias catches himself calling his hotel room "home." "When you call the hotel home," he says ruefully, "you know you're screwed." When he's actually home in Miami, Mondays and Tuesdays are movie nights. He also tools around on the moped Kournikova got him for his May 8 birthday, to replace the Yamaha CR1 motorcycle she gave him for another birthday (he gave it away after flooring it down I-95 at 140 mph once and realizing it was too dangerous). "I'm addicted to speed sometimes," he says, then jokes, "She's trying to kill me."
As for his sleep deprivation, he's considering yoga lessons. "Ugh," he says at the thought. For a man who loves an adrenaline rush, it may be difficult. "I used to write songs when I couldn't sleep," he says. "It was like therapy. That's how I made my albums. But now? Now I just want to go to sleep."
For video from Enrique's photo shoot, go to PEOPLE.COM/ENRIQUEVIDEO