I'm cross-posting my answer from the "Related: using two plastic cups for Kiddush" question:
R' Meir Goldwicht of Yeshiva University addressed this in a Q&A a number of years ago. He felt that doubling a cup does absolutely nothing to solve the problem of the plastic cup not having "Kayamus" as he called it (permanence). He also called into question whether having a plastic cup in the first place was really a problem, since people do refill them and even wash and save them. Since the point of plastic cups is that they could be thrown away, he suggested it was better to use something more permanent, but he did not feel it invalidated the Kiddush to use a plastic cup.