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

Ignorance is bliss. It’s that simple. There are a lot of Americans people no lack the emotional intelligence and maturity to admit when they are wrong. Add to that a lifelong indoctrination (in-school pledge of allegiance, national anthem at sporting events, Hollywood, and country music), and the social stigma of going against the patriotic grain, and you’ve got a recipe for America is the best. It also doesn’t help that schools are mandated to teach a one-sided history that makes America look good.

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?