Now 28, Armstrong is living beyond those boyhood fantasies and his story is the inspirational stuff of a best-selling autobiography and a Hollywood movie in the works.
Just after the 1996 Olympics, America's No. 1 cyclist was diagnosed with testicular cancer so advanced that it had already spread to his lungs and brain. He retreated back to his Austin, Texas, base where doctors gave him a 40 percent chance of survival.
Against those odds he recovered: A testicle amputated, brain tumors removed, lungs bombarded with chemotherapy all formed part of a high-risk aggressive treatment that paid off. Perversely, the ordeal so toughened him mentally and pared him down physically that it helped transform him into a world-beating cyclist.
"Cancer was probably the best thing that ever happened to me. It nearly killed me, but instead it made me a better athlete,'" Armstrong says today.
Before the cancer, Armstrong had been an exciting but erratic prospect on the European-dominated world cycling circuit.
He won the world championships in 1993, but he also was forced to drop out of three of his first four Tours de France, the most prestigious but correspondingly difficult cycling race on earth, because of exhaustion or injury.
His victories on two Tour stages showed how his strength could sometimes prevail. But in the end, the sheer epic scale of the Tour often got the better of his overly broad-shouldered frame.
More important, his tactical grasp of the sport was poor. Through a mix of youthful impatience and misdirected ego, he tried to win rather than stay well positioned within races. He often didn't recognize when to hold back and conserve energy for those decisive moments that the best cyclists recognize and pounce upon.
The chemotherapy winnowed down his bulk, inspiring a more aerodynamic, cycling-friendly shape.
"My whole body's different,'' he said. "That in turn completely changed how I trained, how I ate, how I positioned myself on the bike. And the fact is, I'm 20 pounds lighter than I was in 1996. That's a whole lot of weight when you're trying to climb the Pyrenees or Alps in the middle of the Tour.''
But rebuilding his body required a newfound mental discipline - a willingness to push himself beyond his old limits, and to add maturity and patience to his play book.
"I knew if I could beat cancer, I could get over any mountain,'' he said. Once I got back into racing in 1998, I could always draw strength from the fact that, no matter how hard things might look at a given moment, they could never be as hard as when I was back in Austin in a hospital bed with my hair falling out.''
Armstrong's triumphant return to the Tour in 1999 proved as shocking to some as it was inspiring to others. Many wondered how on earth somebody on his deathbed could miraculously rebound to win one of the world's most grueling sports events.
The suspicion, in a sport already reeling from revelations that top cyclists had abused banned performance-boosting drugs, was that Armstrong was taking chemical shortcuts.
When French media insisted that a steroid-based substance had been found in his urine samples, Armstrong was forced to backtrack on his previous claim to using no inappropriate medicines - but insisted that the offending product was a skin cream for saddle sores.
"I like to talk about cancer, but definitely not about cortisone cream!'' Armstrong jokes today about the embarrassing episode.
Nonetheless, it undercut Armstrong's victory and, combined with accusations he'd defeated a scandal-depleted field, left the survivor feeling he had yet more to prove.
This year's Tour has been, in his words, "a total vindication. It's been sweet.'' After Saturday's stage, he kept his lead of more than six minutes. Now only a ceremonial swing through Paris remains.
"We have to be careful,'' he said. "It's still a stage, it still counts, it's still official, and I'll still not be convinced of victory until we finish the race.''
Armstrong surged ahead of his chief competitors at the first mountain stage, a performance that the top French mountain-climber, Richard Verinque, likened to seeing a plane take off. His lead soon soared to beyond seven minutes, a huge amount in Tour terms.
At the last mountain stage the following week, Armstrong was attacked by Verinque and his two biggest threats: Jan Ullrich of Germany and Marco Pantani of Italy. As he struggled but failed to keep up, Armstrong's legs burned with pain as he suffered through his darkest hour.
But the reborn Armstrong knew his limits - and what it would take to win. He ended the day still in first place and with Pantani resigning the Tour in decisive defeat.
"I knew I was going to crack,'' he recalled. "When I did, I just said: seven minutes is a long time. And this is the lowest point of the Tour de France for me. But if I'm smart and conservative, I won't lose it. So that's why I rode my own tempo and just limited my losses.''
Though Armstrong is a household name in the United States, few actually watch his Tour victories and he knows it. To be appreciated fully by America, he'll have to turn in a winning performance at the Sydney Olympics - and once again the new Lance displays the kind of steely determination that might yet turn Tour yellow into Olympic gold.
"Everybody watches the Olympics,'' he said, quipping that if he doesn't win a medal, people in front of their TVs will complain, "This guy's terrible - that Tour de France must be pretty second rate!''
If anything, Armstrong thinks it's the other way around.
"In the Tour de France, the strongest man wins. The Olympic road race I'm sorry, but it's a lottery. It will be much smarter for me to concentrate my training on the Olympic time trial, which I think I have a good chance to win.''
Time-trial training will also bring the bonus of four-hour workout days rather than his usual seven-hour ones crucial, he thinks, to spending more time on the most rewarding part of his life, his wife Kristin and their 1-year-old boy, Luke.
Fans seeking Armstrong's autograph or picture at the many rural hotels that have been his moving home the past three weeks have been surprised and heartened to see him arm in arm with wife and child. They accompany him rather than stay at the Armstrongs' home in the southern French city of Nice.
"When I'm climbing mountains, I'm thinking of seeing my family at the finish line,'' he said. "I would never even have won one Tour de France, much less two, without Kristin's support. And after my cycling days are done and people forget about my wins, my family's still going to be there for me - and me for them.''