I know making kiddush with a plastic cup is problematic because of requirements. I have seen people using two plastic cups: What is the source for this and is it legitimate?
I'm cross-posting my answer here as well.
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 single plastic cup.
R. Dov Lior answers the question here: http://www.yeshiva.org.il/ask/?id=62394
He says that: When one uses 2 cups, the outer cup is considered a utensil that serves the inner cup, and it will not be thrown away immediately, which perhaps solves the problem of R. Moshe that a single plastic cup is thrown away immediately and is therefore not appropriate for kiddush. However, he says he is not sure this really solves the problem.
Rav Moshe Heinemann answered in a shiur, "Rav Moshe says that using a plastic cup for Kiddush is not lechatchila because it is similar to a kos pogum. Even if you put one plastic cup into a second, that doesn’t help – you’re doubling the problem of having two kos pogums. The reason some people might put one plastic cup into a second one is because they want to avoid getting wet in case there is a crack in one of the cups."