I tried one of those games yesterday. (Sorry, this is getting long.)
Went a bit over the top by choosing a tiny map and allowing 18 civs. Random Civ. Prince difficulty. (I consider myself a monarch player, but tight maps are a bit more difficult.)
Got Isabella (not that great), but a nice coastal town with two fish, two crabs and a whale. Balanced terrain too. There was no space for anyone to found a second town, except the Persians who started near a small group of islands along the coast and got three cities early, and the godless Mongols.
I started well, built some critical wonders, took the lead in the score ranking and collected a huge amount of culture. Reasonable neighbours (Hatty and Roosevelt), Washington got completely surrounded by my culture later, but revolted only once.
I had one bad crisis around 1000 AD: I took an innocent look around the continent with a galley and just saw the Mongol border when Genghis Khan, obviously upset by my reckless provocation, decided to sink my ship. He proceeded by riding through Egypt lands to my country and burning every improvement I had to the ground. I barely hung on, but only because I was so productive to build an archer in one turn (Well, at least at the beginning of the invasion). Ultimately I "won" the war, but crashed down from the top to the middle of the pack.
Soon after that, in a major twist, I sent a caravel + explorer over to the new continent, exactly at 1500 AD (nice timing!). The explorer found a goody hut, and - Behold! - found the Astronomy tech in it. Dumb luck, but it gave me the edge to colonize the northern half of "America", while the tech-leading Persians colonized the South. (BTW the AIs queued up at my palace immediately after I got astronomy to ask for it - and not always in a nice way.)
To avoid another stupid invasion, I then went into submissive mode towards my neighbors, the Eggies and the Persians to get defensive pacts and bribed the Mongols to go hassle someone else (Gandhi. I don't like his Holier-Than-Thou-character.). That worked out nicely. The old continent looked a lot like modern Europe now, with some big countries, a lot of medium civs and some dwarfish states. (Roosevelt controlled exactly two tiles.)
So it became clear that the Persians and me were running away with the game. Cyrus had a sizable tech lead though. But then I pulled off a spectacular move: my midfieldish neighbor Hatty was not only pretty advanced too, but also very sweet to me now (+12) and agreed to engage into a permanent alliance (which lets you share techs and ongoing research.).
I was now ready to cruise to an easy victory. In a retarding (or ********) moment the weak peacemonger Gandhi beat me to the United Nations. I then nearly, but not quite reached a diplomatic victory for our team.
And then, in a shocking and unexpected move, Persia permanently allied with England to take back the lead. Both alliances were now nearly even in points (I had a better score than Cyrus and would have won on my own, but England was slighly bigger than Egypt.) and a mad space race ensued. (The allied Hatty-AI helped nicely and built three or four parts, believe it or not. Another round of applause for the AI programming.)
In the anticlimatic ending, we both missed the last space ship part in 2050 (by four turns in my case, sigh) but I - and by that I mean We - won the game on points. Had the Cyrus/Victoria team not got himself into an extremely silly war with Genghis Khan (I had nothing to do with that at all) shortly before the end, I would have lost.
It was unpredictable and exciting until the end, with major power shifts occurring several times. One of the best I've ever played and proof for the greatness of CIV.