As always, depends on how you define "great". Power? Culture? Influence? Reach? Economy? Durability? Morality? (kind of oxymoronic isn't it)
I'd have to disqualify the Romans. I admire their history and culture yet it would be difficult to say that they were the greatest. They were only the 19th in size in history and to say they ruled most of the known world would be to dismiss China, India, Eastern Europe, and the likes as unknown, uncivilized places. The simple fact is that those parts of the world were certainly known. Whereas Roman battles involved tens of thousands, in China at the same time, there were battles involving hundreds of thousands. So no.
I would have to say the British and the US. They were truly global empires in their reach and one may note that they were both modern empires-empires made possible in large part by modern technology. While ancient and old empires could only go so far in extending their reach, they were literally unlimited. One may also make another note from that fact that they are both the most recent ones. Due to the subjectivity of history, it is only natural to paint the most recent and therefore, the ones that have the most immediate impact on us in contemporary times, as being the greatest. They have much more influence on us which is why we tend to think of them as the "greatest".
I still have a great problem with how a person would define "greatest"