The question here, though, is not what an empire is, although it's important to keep that in mind, but which and what size was the largest empire. Rein Taagepera, who has compiled useful stats, for students, on the duration and size of ancient empires, from 600 B.C. (elsewhere his stats date to 3000 B.C.) to 600 A.D., writes that in the ancient world, the Achaemenid Empire was the largest empire. This doesn't mean it had the most people or lasted longer than others; it just means it was at one time the ancient empire with the largest geographic area. For details on the calculation, you should read the article. At its height the Achaemenid Empire was larger than that of the empire-seizer Alexander the Great:
"A superimposition of the maps of Achaemenid and Alexander's empires shows a 90% match, except that Alexander's realm never reached the peak size of the Achaemenid realm. Alexander was not an empire-founder but an empire-seizer who arrested the decline of the Iranian empire for a few years."
At its greatest extent, in c. 500 B.C., the Achaemenid Empire, under Darius I, was 5.5 square megameters. Just as Alexander did for his empire, so the Achaemenids had earlier taken over the pre-existing Median empire. The Median Empire had reached its peak of 2.8 square megameters in about 585 B.C. -- the largest empire to date, which the Achaemenids took less than a century to almost double.
- "Size and Duration of Empires: Growth-Decline Curves, 600 B.C. to 600 A.D."
Social Science History Vol. 3, 115-138 (1979).
- "The Archaeology of Empires"
Carla M. Sinopoli
Annual Review of Anthropology, Vol. 23 (1994), pp. 159-180.