1. China. From the printing press to gunpowder, how could they be ignored? They win over rome because even during Rome's power China was just as sophisticated in many ways, and even all of Rome's glory really had nothing IMO to compete with the Great Wall.
2. Byzantium/Eastern Roman territory. The eastern section of Rome was more populated with larger cities, more important from a historical point of view (crossroads between the east and west with better access to the silk road), more economical, more sturdy defenses, etc. While western Rome was undeniably important, eastern Rome was even moreso.
3. Rome. In my opinion the only area where western rome "beat" Eastern Rome, at least in terms of importance, is the catholic church. Yes there are orthodox Greeks but not near as many as there are catholics.
4. England/ Great Britain. Dominated the world indisputably for at least 4 centuries, and even in the 20th century their cultural impact with groups such as the Beatles, The Rolling Stones, Led Zepplin, etc was simply huge. Shakespeare gets probably more read than anyone else in the world. Obviously their influence has been well on the decline for years in more ways than one, but this isn't only about current times, it's about history overall.
5. United States. We've been at the top of the pack since the end of ww2. We were the tiebreakers in both ww1 and ww2 and we have the manpower to pretty much single handedly control middle eastern affairs. I would have actually ranked us lower on the list, but with Yahoo, eBay, Amazon, Google, Apple, Microsoft, Facebook and the internet itself all being American inventions, I'm going to have to at least put the United States in the top 5.
5. France. France and Spain were the closest colonial competitors to the UK. France wasn't occupied for near as long as Spain was by foreign invaders, and France also has had far more of a cultural impact, so they get a higher ranking than Spain.
6. Spain. The only thing I will say for Spain is that they got much more gold out of their conquests than France did because of where they conquered.
7. Persia. The reason Persia got as big as it was and for so long (I include both the "classic" persia and Iran from that time to where it got invaded by Arabs as all the same) was because of it's stability and tolerance of those they conquered. This they managed better than the Romans, allowing them to compete with them indefinitely whereas the rest of the (known) civilized world fell. Persians influenced Judaism which in turn influenced Christianity and even Islam.
8. Arabia. Founders of Islam, the second largest religion in the world. Countless contributions such as the ones they made in medicine and the invention of Algebra. Need I say more?
9. India. Religiously speaking India has been more prominent than any other country in the world. That alone earns them a spot on the top 10 IMO.
10. Russia. Largest country in the world, both Napolean and Hitler couldn't get it, Rivals with United States after ww2, helped launch the space race. Need I say more?
11. Germany/Prussia. Huge military culture which made the dominate Europe for some time. I'd go as far to say that they had a lot to do with the fall of Europe's colonization. England and France couldn't have troops overseas when Germany/Prussia was impolitely knocking on the door at home.
12. Greece. They are more or less the birthplace of western civilization, but they didn't have near the long-lasting effects of Rome, much less eastern Rome. Still they conquered lots of territory in a short period of time and expanded Greek culture around the world. Many philosophers and such came from Greece.
13. Ottomans/Turkey. Brought down the mighty Byzantine empire which says something in itself. They had enemies on all sides, and kicked their butt on all sides until losing out in ww1.7. Egypt.
14. Japan. I admit my knowledge of Japan is pretty limited, but still one of the more influential countries. I would have actually ranked them lower, but they had a profound influence on WW2, much more so than the dutch, thus giving them barely enough edge to secure a higher position on the list.
15. Netherlands. The dutch colonies were not as great as that of England/France/Spain, but they were arguably number four.
16. Portugal. The smallest significant European colonial power. Were it not for them Brazil would have been much different, so yeah, they're obviously important for having influence on such a big place.
17. Mongolia. Honestly it was a tossup between Portugal and Mongolia, but Portugal historically had less poverty. Even Portugal today has more influence than Mongolia internationally speaking, thus giving them the higher position. However the Mongolian empire was the largest empire in the world of a single stretch of land. It didn't last very long, but still a very impressive thing to accomplish.
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The others on the list I don't know enough about to have an educated opinion.