Lose the sprint, win the marathon

Chris has just completed the Tour de France 2015, after 21 grueling races. There were 198 riders in the Tour de France competition. After the first few days, Chris was always showing signs of hopelessness. Coincidentally, he ranked \(n\)-th in each individual race. However, much to his astonishment, he ended up taking home a gold medal because, despite his seemingly lackluster performance, he still managed to finish with the fastest combined time!

What is the maximum possible value of \(n\)?

Image Credit: Flickr Peter Mooney.
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