The Greatest Empire?

[Storebought]

For longevity and breadth, I'd go along with China. Imperial government, with interregnums, lasted 2000 years in China, whereas, even including Byzantium as the legitimate legal successor of the Roman Empire (which I think is misleading after Justinian), Rome/Byzantium lasted "only" 1400 years, the last few centuries as an isolated impoverished city-state.

As far as cultural legacies go, the Roman legacy, outside of the Latin language, was literally forgotten in the west for 800 years after the fall of the western capital, and had to be sought-out and self-consciously revived by Italian lawyers and antiquarians. In China, the cultural legacy sustained in the empire had to be self-consciously forgotten to move past its collapse.

Not to mention, Rome at its height governed 50 million subjects; China and its vassals consistently governed 1/5 of all humans on the planet.

But that is all ancient history.

As far as "modern" post-Columbus colonial empires go, then, obviously, the British Empire is by far the most encompassing and audacious ... in 19th century India alone, ~60,000 British troops and 1000 civil servants effectively occupied a nation of 200 million. And the British Empire was profitable -- in 400 years, the Romans heavily exploited their imperial possessions, but never once managed to make them all profitable.

I'd give the Spanish and the Russian Empires a mention as well for ruthlessness and shamelessness.