As of Gen VI, the starting Pokemon who learn the most moves are:
First, I just want to make it clear I counted all these manually and that I might have a miscount somewhere. You can check, but you probably don't have time for that. They're pretty much all in the same general ballpark with Chespin leading all and Fennekin at the bottom. I counted by actual starting Pokemon, not their evolutions. Also, I didn't count repeats.