They are not strictly speaking civilisations - well certainly not ones that passed the test of time with the exception of the Ptolomies but I find the wars of the diadochi era interesting.
That period is the battles to hold Alexanders empire in which his generals fought as per his (possible) last words when asked who should rule his vast empire "the strongest".
I like the Khmer empire the best, they had great architecture. But they are from around the Mongol time, so I guess it's a little late too.
My second is the Indus Valley Civilization.
k how about we extend this to middle ages. In that case, my top 5 will be, from 5 to 1, ancient greece, mongols, France, tie with Holy roman empire and habsburgs, and rome.
By what standards are we judging a civilization as "best"? If time is a factor, I nominate the Chinese. Their civilization has existed for over 5,000 years without disappearing like most of the rest.
I have 3, and rightly so.. Wu Wei and Shu, because you can't talk about one without talking about the other two. Reasoning: 2-way civil wars are cool, but 3-way civil wars are downright fantastic to read about.
For the middle ages, I would have to say all of Western Europe was awesome. I also liked the Aztec Triple Alliance. Some people don't like them because they fell so easily to the Spanish, but I think they could have survived if all of their city-states united.