Definition of "largest" for the purpose of this question: the greatest number of flight operations per year on average.
What is the largest flight school in the United States that does not offer a degree?
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According to Wikipedia, "ATP Flight School ... is the largest flight training company in the United States."
However, volume does not necessarily mean quality, and the focus of ATP and other big-name flight schools seems to be on students who want an airline career—and pushing them through the training as quickly (and profitably) as they can. If that doesn't meet your goals, or you don't have large amounts of both time and money available to dedicate to it, you are probably better served going another route.