You'd never guess from meeting him on vacation in Dubai, still less from watching him play tennis, but Roger Federer, at 25, has a problem. The number one player in the world for more than three years is currently forced to ask himself a question that few have ever had to confront: How do you reveal the perfection of your game when there is no one around to push you to your limits? The very greatest have always been shown in a rivalry worthy of the name: Ali had Frazier, Sampras had Agassi, Woods is lucky to now have Mickelson. Such is Federer's domination of tennis, however, that he seems increasingly desperate to discover a nemesis. Legends, as he knows, are not created by statistics.

[See a slideshow of Annie Leibovitz's photographs of Federer in Dubai.]

For a while now, as a result, he has looked for competition from beyond the game. (In the past three years, the Spaniard Rafael Nadal dealt him annual humblings, but almost exclusively on clay.) Federer has befriended Tiger Woods, one of the few sportsmen who have shared his preeminence, and pretends a rivalry with him. Failing that, he tries to force himself to play against the history books, or to look across the net at immortality.

Last fall, as Federer was settling in to his new apartment at the vast under-construction marina in Dubai, he heard that the greatest of all tennis head-to-heads was being reprised up the road: Björn Borg was billed to play John McEnroe in the "Legends Rock Dubai" exhibition event. Though they played out their indelible rivalry before Federer was born, he has always been a Borg man (everyone is one or the other). He and the iconic Swede had spoken on the phone, but they had never properly met.

Federer retains an aura of boyishness, and in the corner of a bar in one of Dubai's decadently opulent, apparently gold-plated hotels, he giggles a bit as he recalls how he approached Borg. "I called this guy who knew him and said, 'Tell Björn I will be on court tomorrow at 10.' I wanted to say something specific so it would be easy for him to say no. I didn't want to embarrass him."

Federer need not have worried. The holder of 10 Grand Slam titles (and counting) arrived at the appointed hour to find the 11-time Grand Slam winner already warming up. "He played very well," Federer recalls, with an element of awe. "I could easily still practice with him, you know."