"How many marijuana plants can you grow if you become a certified medical marijuana patient in New York?"
Under New York state law, patients are not authorized to grow their own cannabis. Growing the plant is strictly regulated by New York State and only licensed providers can grow, cultivate, and transport. I would not try to test the system in New York with your own plants, as it is a highly regulated market with strict penalties.
Under New York law, growing marijuana at home or in any unlicensed environment is against the law. The only place to legally obtain medical cannabis is at a licensed dispensary.