This question could mean a couple of things. The two things that first come to mind are very different. One is a more personal and broad question, while the other is very much definite in nature. First is if a person ever stops chasing the American Dream. The second is if there will ever be a day where the American Dream ceases to exist, and what may cause that.
First, I do not think there is an end to the American Dream. I believe that as life goes on you change your goals based on where you are in life. Even though you may have all the things done you set out to do years ago, there is always a goal to be had–whether you realize it or not. Everything you do is a step to make it to tomorrow, and that alone is a goal we all have. Dreams are ever changing and may end up completely different than when it began. Either way, the dream will live on till you pass.
Secondly, I do think the American Dream could fade away one day. Assuming American does not outlive the end times, I think something may happen that causes America to possibly dissipate. After that, the generations of people who grew up in America will fade out to where nobody is left. How is there a dream with no country it represents? Another idea is that out country changes to a point where freedoms become stripped from us and causes a life with no goals, but only set guidelines and rules. The last option would be some utter destruction of America that also destroys the dream. Otherwise, I think the idea of the dream will keep changing with generations and mold into something new for everyone that follows me.
I think the dream will not ever come to an end in either fashion. While it may dissipate later on in life, As long as I live I will have some variation of my American Dream that I continue to strive for.