Has to be deity - let's look at some (simplified) facts.
History shows that whatever your empire level, it will fall.
Starting at 4,000 BC - and playing a little fast and loose with dates. Don't pull me up on the factoids here, I'm just establishing a principle and dealing with a handful of major historical players.
The Phoenicians
- don't know much about them, but their ships carried them all around Europe. Excellent trade, excellent geographical knowledge of their area, knew how to mine tin etc. But as I understand it, natural disasters finished them off. They were a great civ - all gone.
Eliminated by natural disaster then.
The Egyptians
- the great pyramid, if what 've heard is correct, contains more stone than the combined cathedrals of Europe in the middle ages. No shortage of workers then. Good for religion, good for military power. But they were done by the Romans - never a major force again.
Eliminated by enemy might.
The Greeks
- Fathers of modern philosophy, fathers of modern democracy, quite powerful militarily. But with fractious city states and Sparta on one side and Italy on the other, they were gobbled up by Rome.
Eliminated by might, sapped by petty wars.
The Romans
- An awesome civ, no need to mention their many accomplishments.
Wiped out by barbarian hordes, coupled with the strain of keeping hold of such a large empire (consider it like a city in CivIII flipping over to a different civ?)
Eliminated by barbarians, eventually.
The British Empire
- It is estimated that in the 1800s, a quarter of the world's population was in the British Empire. Could they have kept it, given a few twists of fate? But they 'lost' India, gave independence to many other civs as well, new political and humanitarian theories discouraged the idea of 'empire', and two world wars saw them go from #1 to lower top-ten in fifty-odd years.
Their maritime might was unsurpassed for hundreds of years in a European arena where Macchiavellian tactics and attempted conquest was the norm. There were military alliances all over the place and everyone was scrabbling for territory all over the world.
- eliminated by a combination of factors, city (country) flipping back to former ownership, political factors, and two titanic wars knocked the stuffing out of them.
So history shows that the major players I've picked out here (and there are many others e.g. Soviet Russia which were massive and couldn't last for a century) , with many unique attributes and strengths, did not have what it takes to play a winning game of Civ3.
- that to me spells 'deity' level.
) rather than computers at this time, that accounts for why things don't generally happen the way they do in an average single-player civ game. 
