Nice game entropie.
I personally think that Alexander the Great's Hellenistic Empire was probably the best in the world. You could argue Napoleon, but his reign was short and came to a abrupt end in about 1815 thanks to Typhoid fever and the harsh weather conditions of the Eastern-Russian planes.
Furthermore, the Persian Empire, in about 500 BC proves awful powerful in light of Empires throughout our world's history. Conquering the entire Aegean peninsula after their famous, or infamous (depends on your perspective) over the Spartans at Thermophile.
And lastly: the Roman Empire. One could argue that the Roman Empire can be considered to be the strongest in history, but when you compare it to the domination and imperialistic nature of the Hellenistic Period, the Roman Empire proves far less victorious. Conquering only about half of Europe (as compared to half of Europe, and almost half of Asia under Alexander the Great), the Roman Empire was only successful because the Romans faced hardly any an opposing foe..it was all about time. Poor leadership, over population, an imbalanced economy..all of these things lead to the downfall in 476. It's not like it was a surprise.
And, oh, I almost forgot. The Mongol Empire. Arguably one of the most tactical, war-hungry, victorious, and mutant ways of lifestyle, great leaders such as Genghis Khan and Kublai Khan lead their hordes of power hungry, horse backed, archers through the Gobi desert managing to conquer China, hold Mongolia, and invading parts of India, Siberia, and parts of the western Middle-East. However, as these maneuvers were less of a governmental-fete (as no clear, definite, or efficient faucet of government held this band wagon together), the Mongolian Empire just misses the position of best Empire in the world.
When you consider the modern-era, and thanks to our friend Imperialism, The British Empire must be considered. Colonizing posts around the world, the brave and powerful Royal Navy is responsible for the abduction of many a native land into the British Reign of Power. Look at India, The United States...both free nations today, but not after a complete social, political, and economic revolution. It is a blessing to have such an influential figure as Ghandi to back up Indian beliefs, spark patriotism, and lead to the freeing of the bonds of Britain. It is remarkable.
In short, we learned that Empires can be conquering, powerful, selfish, ravage, and can conquer up to a third of the world. But ultimately, it comes down to the tactics at which these actions were carried out, the leaders in charge, and the effectiveness of the governmental infrastructure that followed. It can easily be said that Alexander the Great's Empire best meets this criteria, so let us take a moment to honor all of those many soldiers that gave their lives and their time to the creation of the World's Largest Empire.
You forgot the russian empire, china, otttoman empire and the arabian empire. Also the japanese empire and nazi germany was quite powerful in world war 2.