First, the Roman Republic had a very strong and orderly army. It had so many rules from how to make a battle formation to how to make camp. It was said that people in neighboring areas ware often surprised and feared Roman soldiers for their orderliness and bravery. In fact, most of the territory of Ancient Rome was gained when it was republican constitution, not the imperial constitution. For example, The Italian Peninsula, Iberian Peninsula, Sicilia, and so on, were gained when Rome was a Republic.
The second reason why I think the Roman Republic was the greatest empire is that it had a good constitution. Many people could take part in politics and they had chances to make a success of life. Its constitution consisted of the Senate (assembly of the noble class of people), the Comitia (assembly of citizens) and some magistrates (elected officials). The Comitia had strong power to be able to make a law without the Senate. All Roman citizens could magistrates by election, regardless of their birth. What’s more, a citizen with experience being a magistrate could get qualifications to join the Senate. In this empire the door of politics was open to anyone. I think suffrage is very important for a nation.
To conclude, the Roman Republic was quite excellent in strength and political constitution. It was a republic from 509 B.C. to 27 B.C., then it became the Roman Empire, and the eastern half (the Byzantine Empire) continued till 1453 A.D. I think it is surprising that a small city state expanded so largely and continued so long. So this is a suitable empire to call the greatest empire.