Egypt
India
China
Babylon
Greece
Persia
Carthage
Rome
Japan
Ethiopia
Maya
Vikings
Arabia
Khmer
Spain
France
England
Germany
Russia
Dutch
Mali
Portugal
Inca
Mongols
Aztecs
Turkey
America
Cut Aztecs and Inca, because they are just booped.
Cut Mali, Ethiopia, Carthage, Egypt, and India because they can just not keep up with the competition.
Cut Khmer, Vikings, Maya, Portugal, and Dutch, because, while they may have the potential to become powerful, they are nothing close to powerhouses.
That Leaves:
China: By far one of the most powerful until the need to expand during the industrial era distabalizes it. However, the lead that it can have is amazing.
Babylon: While it can become a giga-power, it has far too many little things to worry about for it to be played optimumly, and luck is very important.
Greece: Has potential to out-tech everyone to a ridiculous extreme, but can not stay stable past the Renaissance.
Persia: It just keeps expanding forever, but it has tons to worry about.
Rome: The best in my opinion, if it survives the barbarians while keeping a tech lead, it
can be as powerful as China and as technological as Greece, while still being able to expand stably. Tons to worry about, but nothing that really poses a threat. More military is needed to keep the cities happy then to defend them.
Japan: Nothing to really worry about but the plague, amazing research, and what it lacks, it can take from China.
Arabia: Amazing start in 600, but almost as much to worry about as Babylon.
Spain: Can expand all over the wonderful land in America, but it can just not keep up with the rest of Europe tech-wise.
France: The tech leader of Europe for a while, and that is saying something. Has great potential to dominate North America and/or Europe if it chooses.
England: Ridiculous tech rate and territory to expand too, it is only limited by its inability to stably hold great land other than itself and N. America.
Germany: Probably the best production civ, with insane UP and UU, but stability can haunt you, even if pazers can recapture most of Europe in a few turns.
Russia: Endless territory to expand to, and can survive through Europe as well, but horrible foreign relations give you war when you least want it. As if it would make a difference with the winter and insane production.
Mongols: They can expand into China's amazing land, and all the way into Europe, but their starting units are not enough to make up for horrible stability. Or get past a well-defended city.
Turkey: Everything seems against it, but the only civ other than Rome that can stably get all of the great land in both Europe and the Middle East can become extremely powerful extremely quickly, and stay that way.
America: If only it spawned earlier... With a free continent with land as good as Europe to expand through, easily and stably, it is sadly gimped by its late start.
I would say that it depends mostly on stability potential, starting land and technologies, and potential to expand.
Generally speaking, if you can keep and hold Kiev with the civ, it will own. Denver and Chicago are nice too. All other really good cities have insane competition.
Cut Babylon, Spain, and Mongols, as they just barely made the previous cut.
China
Greece
Persia
Rome
Japan
Arabia
France
England
Germany
Russia
Turkey
America
"Powerful Civs"
Ancient: Get an dearly wonder and tech lead, must be able to compete with the ridiculous list of techs that the classical civs spawn with.
Classical: Must beat out the other classical civs, and survive the barbarian invasions, while meeting Arabia's great tech start. Spam wonders.
Medieval: Tech to certain techs quickly, nab much of Europe.
Renaissance: Beeline to get the conquerors even, Liberalism, and Astronomy to start out colonization.
Industrial: Keep up with lightening expansion and teching without becoming unstable. This is where most older civs decline.
Modern: You should be by far the most powerful by now. Crush all those that oppose you, and reclaim your empire from civil war in a few turns if you have to. Choose your victory.
Classical
China
Greece
Persia
Rome
Japan
Medieval
Arabia
France
England
Germany
Russia
Later
Turkey
America
While the medieval and later civs may be able to establish themselves, and become superpowers, they can never make up for the lead that the classical civs could have had before they were born.
While the classical civs may have an advantage over the medieval and later civs to begin with, they can not keep up with the continuous expansion of them far into modern times.
Humorously, Arabia practically counts as an older civ by this definition.
My opinion:
Rome
[a mess of indecisiveness]
tl;dr
Is it better to have a lead and hold it, or continuously grow? Well, Rome can do both, so yeah.