Yes, most Americans do genuinely believe this is the greatest country on earth. While that opinion may not be entirely objective, I don't see how it could be said to be untrue. It's merely an opinion. It should be understood that America is made up of an extremely diverse assortment of people. Our sense of national identity is what binds us all into one relatively cohesive society. When Americans talk about their country, they are really talking about their culture which is based on a common set of values established by our forefathers, and refined by successive generations. So when an American says, "this is the greatest country on earth", he means "my culture is superior to all others". This, in fact, is a belief common to all people everywhere. Most people are not bold enough to come right out and admit they think that, but it's just human nature.