The USA is among the historic greats (the UK, the Byzantine Empire, etc), but it's a bit difficult to say we're currently the greatest country in the world. We're certainly the greatest military force in the world, and our global influence on culture is unmatched.
But what a weird question that is, when you think about it. A country like Denmark, for example, isn't concerned with being the "greatest country" in the world, as if it's a contest, but it certainly takes measures to sustain and increase the standard of living of its people (which the USA does not seem to care as much about).