You can't really stockpile missiles, their explosives and fuel needs to be replaced with some interval, so having a massive stockpile doesn't make that much sense, because it's as you suggests expensive to maintain.

In the case of an war (like you know, a war-war) the US is perfectly capable of ramping up manufacturing to ungodly numbers (but it won't be cheap). I would however like to see what Ukraine does, if they get the go-ahead to start making Patriot missiles. They probably won't just do a run of 50 or 100 missiles and call it a day.