Given the definition of "greatness" I'm getting from this thread, the American Empire is clearly the greatest.
First, though, we need to establish that America is an empire. An empire typically encompasses multiple "nations." The USA proper is not an empire at this point. Most everyone everywhere in the US from Alaska to Florida regard themselves as nationally American. However, the US does have foreign possessions that include groups that do not count themselves as American: American Samoa, Puerto Rico, various island groups, etc.
Now why is the USA the "greatest" empire? Let's count the qualifications for greatness as: Size, Military Power, Economic Power, Cultural Power, and Authority (authority: the ability to get what you want done by others or to be believed by others simply by saying so, rather than direct use of force.) Let's take the US at it's high point to date: around 1999. The United States clearly fails in size. It never constructed the huge sort of empire that the European powers could because it was too young during the main bursts of colonisation. However, it is quite large (currently the 3rd largest country in the world by land mass, as well as in terms of population.) The United States more makes up for that shortcoming in military power. The USA has the most advanced military in the world in proportion to all other militaries. This is significant because while the British armed forces, Russian Imperial forces (in the early 1700s), Ottoman and Roman forces were all more advanced than other states', they were still relatively close in terms of technological advances. The USA, though, has military technology far outstripping its next closest competitors (probably the UK, France, and just maybe Russia.) The US Navy is capable of projecting power far inland, and, much as the British Empire's once did, is widely credited with ensuring the security of the world's waterways, giving the US extraordinary direct hard power over the global economy. Furthermore, the USA in 1999 had a virtually monopoly on space. Finally, the US has more serviceable nukes than anyone. If worst comes to worst, America can blow any nation off the face of the earth and if it doesn't have nukes, the likelihood of retaliation in kind by a nuclear power is virtually nil. Now on to economic power: the United States in 1999 (and still mostly does) comprised the global economic infrastructure. The US bought everything from everyone, our dollar was the international currency, and we had (and have) the largest and most powerful economy there is. In terms of cultural or soft power, the United States has Hollywood, most of the major media services, is effectively the "home" of the internet, had enormous respect abroad thanks to the triumph over the Soviet Empire, is the largest English speaking nation, possesses the worlds largest multinational corporations, and has the best universities in the world. Now for authority. The USA could get pretty much whatever it wanted done just by asking. The US's military, economic, and diplomatic prowess were unmatched and the States were widely viewed a "positive force" in the world.
The US simply was (and still is) the closest thing the world has ever had to a global hegemon. And certainly still is a regional hegemon (no offence meant to the Canadians.) And that position was further reaffirmed and entrenched by something relatively new: international law. International law, by my view, doesn't matter for much, but it does have some importance, and the vast majority of world institutions were made by or with the USA and thrived thanks to America's support, this resulted in institutions that favour the status quo of US dominance. (Which is why George Bush's insistence on ignoring international institutions and norms to pursue some sort of dogmatic independent-minded nationalism is arguably the most idiotic thing he could do.)
So in the end, the USA in 1999 is the resounding choice for all possible assessments of "greatness," except for size which in the modern day is both an obsolete measure and which is made up for by just how far ahead America is in all the other categories.
PS- The level of mindless nationalism, hatred and inability to view past events through a historical lens is extraordinary. I think it can be agreed that the British Empire was worse for the world than the US Empire. It had a lot longer to screw things up, and it existed in a time when overt colonialism was totally acceptable and not only politically palatable, but in fact desirable. All Western (and most non-western) cultures avidly pursued empire, as such it is entirely unreasonable to pass judgment on these empires based on modern perceptions and morals. American territory would've been imperialised by someone. Africa would have been dominated by one group or another. Debating the legitimacy of past action based on modern standards will do nothing to promote the stability and prosperity of the modern political situation.
			
			First, though, we need to establish that America is an empire. An empire typically encompasses multiple "nations." The USA proper is not an empire at this point. Most everyone everywhere in the US from Alaska to Florida regard themselves as nationally American. However, the US does have foreign possessions that include groups that do not count themselves as American: American Samoa, Puerto Rico, various island groups, etc.
Now why is the USA the "greatest" empire? Let's count the qualifications for greatness as: Size, Military Power, Economic Power, Cultural Power, and Authority (authority: the ability to get what you want done by others or to be believed by others simply by saying so, rather than direct use of force.) Let's take the US at it's high point to date: around 1999. The United States clearly fails in size. It never constructed the huge sort of empire that the European powers could because it was too young during the main bursts of colonisation. However, it is quite large (currently the 3rd largest country in the world by land mass, as well as in terms of population.) The United States more makes up for that shortcoming in military power. The USA has the most advanced military in the world in proportion to all other militaries. This is significant because while the British armed forces, Russian Imperial forces (in the early 1700s), Ottoman and Roman forces were all more advanced than other states', they were still relatively close in terms of technological advances. The USA, though, has military technology far outstripping its next closest competitors (probably the UK, France, and just maybe Russia.) The US Navy is capable of projecting power far inland, and, much as the British Empire's once did, is widely credited with ensuring the security of the world's waterways, giving the US extraordinary direct hard power over the global economy. Furthermore, the USA in 1999 had a virtually monopoly on space. Finally, the US has more serviceable nukes than anyone. If worst comes to worst, America can blow any nation off the face of the earth and if it doesn't have nukes, the likelihood of retaliation in kind by a nuclear power is virtually nil. Now on to economic power: the United States in 1999 (and still mostly does) comprised the global economic infrastructure. The US bought everything from everyone, our dollar was the international currency, and we had (and have) the largest and most powerful economy there is. In terms of cultural or soft power, the United States has Hollywood, most of the major media services, is effectively the "home" of the internet, had enormous respect abroad thanks to the triumph over the Soviet Empire, is the largest English speaking nation, possesses the worlds largest multinational corporations, and has the best universities in the world. Now for authority. The USA could get pretty much whatever it wanted done just by asking. The US's military, economic, and diplomatic prowess were unmatched and the States were widely viewed a "positive force" in the world.
The US simply was (and still is) the closest thing the world has ever had to a global hegemon. And certainly still is a regional hegemon (no offence meant to the Canadians.) And that position was further reaffirmed and entrenched by something relatively new: international law. International law, by my view, doesn't matter for much, but it does have some importance, and the vast majority of world institutions were made by or with the USA and thrived thanks to America's support, this resulted in institutions that favour the status quo of US dominance. (Which is why George Bush's insistence on ignoring international institutions and norms to pursue some sort of dogmatic independent-minded nationalism is arguably the most idiotic thing he could do.)
So in the end, the USA in 1999 is the resounding choice for all possible assessments of "greatness," except for size which in the modern day is both an obsolete measure and which is made up for by just how far ahead America is in all the other categories.
PS- The level of mindless nationalism, hatred and inability to view past events through a historical lens is extraordinary. I think it can be agreed that the British Empire was worse for the world than the US Empire. It had a lot longer to screw things up, and it existed in a time when overt colonialism was totally acceptable and not only politically palatable, but in fact desirable. All Western (and most non-western) cultures avidly pursued empire, as such it is entirely unreasonable to pass judgment on these empires based on modern perceptions and morals. American territory would've been imperialised by someone. Africa would have been dominated by one group or another. Debating the legitimacy of past action based on modern standards will do nothing to promote the stability and prosperity of the modern political situation.
 
			 
 
		 
 
		 
 
		 
 
		 
 
		