PUMaC 2014 Combinatrics A Problem 2

Assume you have a magical pizza in the shape of an infinite plane. You have a magical pizza cutter that can cut in the shape of an infinite line, but it can only be used 14 times. To share with as many of your friends as possible, you cut the pizza in a way that maximizes the number of finite pieces (the infinite pieces have infinite mass, so you can’t lift them up). How many finite pieces of pizza do you have?