The United States of America have been around, in some form or another, since the 1500s. That's really not so bad, in terms of historical legacy. Granted, back in the 1500s, what existed was mostly just European colonists and native American tribal nations (who had a fascinating history prior to their genocide).
That said, I agree completely with Akka. Greece, Egypt, Persia, Babylon/Sumeria, and Rome, would be my choices. India and China desperately belong on that list, too, but I couldn't fit them. Maybe China instead of Rome.
The Sumerians were the first civilized peoples of which we know. The Babylonians -- their successors -- are responsible for much of what we take for granted today. As has been mentioned, they held the number 12 to be sacred and mystical, thus 60 minute hours and 24 hour days. We also have 28 day lunar cycles. Astrology is Babylonian in origin. The Babylonians feature heavily in Jewish history, as well.
The Egyptians made a strong, lasting empire out of two rivers. Out of the millennia that they existed during, they enslaved many people, built several wonders -- including the only ancient wonder that is still standing today -- and conquered many nations. They are responsible for the invention of many scientific and cultural ideas. There would be no written language without Egyptian hieroglyphics, and there would be no Internet without written language.
In reality, the Greeks were just Egyptians who were more philosophical. The Greek pantheon is simply the Egyptian pantheon with different names, then filled in with some new deities and events. The Greek civilization grew powerful in philosophy, science, culture and military might, eclipsing any Egyptian origin. Pi, democracy, modern philosophy, and modern architecture are all Greek in origin. It all started in Greece, baby.
Rome is where Europe was birthed. In another sense, you could say Germany or France. I would agree with you. However, to go back to its furthest roots, you have to go to Rome. Rome conquered all of Europe, standardized the language, and made everyone one religion. That's an accomplishment. I'm not especially enamored of Rome, but I admire their professional army. In all other respects, the Romans were simply Greeks with a different haircut.
Until Alexander the Great conquered Persia, it was a thriving empire. Cyrus the Great, the founder of Persia, was probably the first enlightened monarch of the classical world! Cyrus freed all the slaves in Babylon, instituted the first human rights laws, and, generally, kicked some serious ass when he went to war. Later leaders, such as Darius I, continued this enlightened rule, but Xerxes probably deserved what he got from Alexander. Alas, but all great empires must come to an end, and Persia's time had come. From Persia, we get humanitarian philosophical ideals, math, science, Zoroastrianism, and a strong cultural heritage that still exists today (scimitars, Ali Baba, genies, etc).
England is just Romans on an island. France is just Gauls and Romans with an attitude problem (though I love France). Germany has more of a claim to be in the top 10 than most other European nations: English is just a German dialect, the Germans are famous engineers, and the German war machine is legendary. German tribes sacked mighty Rome. German mercenaries participated in and tipped many historical wars. German scientists and philosophers have historically changed entire European thought patterns (Nietzsche, Freud, Marx, Engels, etc). How can one say that a bunch of Romans on an Island compare to this?
The U.S., for all of its greatness, is just a bunch of Englishmen on an even bigger island. BFD.
China/Asia is so influencial, I wouldn't even know where to start. However, given my own acknowledged Indo-Eurocentrism, I knocked them down to "runner-up" status. I love Asian history, but I can't really relate to it like I can Indo-European history.