The Greatest Empire?

[Frodo]

Well, I would argue that the USA is an empire in the sense that it has "colonies", i.e. territories that don't have full rights as citizens. In fact, even DC gets no votes in Congress.

If we are an empire, we are made distinct by the fact that our empire has virtually no colonies (except the few scattered islands of no importance). 

[Frodo]

I would argue that Achaemenid Persia is the greatest empire ever.  At its peak slightly over half the human population in the world was within the borders of Achaemenid Persia.  This is a record that has never been exceeded by any other political entity. 

Surely the Mongol Empire dwarfed all others in its day, at least when it came to sheer land-mass: