Why do some people from United States still believe that their country is the greatest in the world?

I have no idea what "greatest" means. It's a superlative word, but implies no meaningful measurement that can be tested and verified.

Also the word "believe" is problematic. "think" might be better. Or "hold the opinion".

Both of these points support the idea that you're asking about an emotional state, rather than a intellectual position. Basically you're asking "why do people like the place that they live and mindlessly cheer for it when sociologically appropriate moments for such displays become available". That's a question that answers itself, is it not?