Is America the Greatest Country in the World?
Let's try to remove the patriotism from this answer
If you were to look at the United States in the context of the history of the modern world (year 300AD+), the U.S. is conducting a massive social government experiment. Never has a country lasted long-term (250 years plus) when its government consists solely of individuals chosen by the citizens of that country. The U.S. was the first to do this. As a result of that and of certain freedoms and powers granted to the citizens of the country that no other country offered until U.S. came along, no other country in the history of the world has experienced such an immense entrepreneurial spirit.
Thanks in part to government's hands off approach (low taxes and tariffs, trade laws across states, etc.), the 1800s and early 1900s saw the largest industrial revolution ever, and a tiny, brand new country explode into the largest economic power. Historically, the U.S. was the ONLY place a person could actually work themselves up the ladder of success. A person didn't have to come from, marry into, or form a relationship with a wealthy or influential family/individual in order to become wealthy themselves. They could actually generate their own wealth and success. This was the first country ever to allow such opportunity on a wide-scale. (Yes, there were others who accomplished this, but they were extremely rare.)
Also thanks to freedom of press and thanks to the government not deciding what is popular, in demand, etc. artists flourished. It took many, many years, but artists, writers, musicians, architects, etc. in the U.S. finally started having their own voice and now the U.S. houses the biggest entertainment industry. Many such examples can be given that result from the principles established in our Constitution. For that reason, many people consider the U.S. the greatest country in the world - people from other countries, too.
An interesting observation is now that many other countries have taken on various qualities, and even laws, of the U.S., they are becoming more successful and their citizens are experiencing a standard of living comparable to that of the average U.S. citizen, and are even afforded the same opportunities for financial and creative success. Thus is one reason why citizens of other countries can now claim to be the "greatest" while providing solid explanations as to why.
One thing that's concerning, however, is that government now plays a bigger and bigger part of the average citizen's lives. Business laws are stifling. The media has become even more powerful and are able to shut down ideas and big solutions. (How many of you have heard about the latest Climate-gate, where pro-Global Warming scientists and cheerleaders have essentially not allowed any scientist to present their findings that global warming is NOT man-made? If you haven't, blame Obama-democrats and the Obama-loving media.) Similarly, big business - or I should say, big money - has suppressed some peoples' ideas, and thus their success.
And so, as a result of government (both dems and reps) and the media, the U.S. is on a downward spiral. Other countries may soon be able to take the "greatest country" trophy the U.S. has held onto for so long.
In order to answer this question, one must get beyond patriotism and look at things such as opportunity for success, personal freedoms, standard of living, and so on. With those in mind, it's hard for others to compete with the U.S.