The "American Dream" seems to usually be defined exclusively in terms of material wealth, which is a little troubling, but is at the same time very, very American. So I will roll with that definition.
That said, I don't think of the American Dream as necessarily greedy or outrageous. To me, it implies only a modest, comfortable standard of living. Things like: Owning your own home. Two cars in the garage. A job that carries health benefits and a 401(k). Three decent meals a day. The financial means to raise kids and send them to college. A little left over at the end of the day for the occasional vacation: if not to Europe, then at least to Florida.
However, because of changing economic conditions, the American Dream is feeling out of reach for a lot of people these days. I think this harsh reality hits the young especially: Crushing student debt, a lack of decent jobs that match their degrees, and home prices that have risen far faster than pay rises have kept up.
All this in spite of low unemployment and an economy that, on paper, is doing very well right. Indeed, I don't deny this economy is working well for a lot of folks, but only if you're already on a certain "track" that many aren't.
This is why a lot of young people are gravitating toward left-wing candidates like Bernie Sanders. They're not doing it because they're brainwashed or they fell in love with Karl Marx at the university. Rather, they're motivated by hard-nosed, self-interested economic realities. Example: When Sanders promises to cancel all student debt (a policy I personally disagree with, and which is very, very likely never going to happen, even if he's elected), Sanders' young supporters hear a huge boon to their finances, and a head-start on the down-payment they hope to be able to one day afford.
Final thought for now: What we consider to be living modestly and comfortably here in America is actually extremely wealthy and decadent to the vast majority of humanity.
Through it all, we should try to count our blessings more often than we do.