To me, playing Civ isn't just about winning the game's own conditions - it's about setting a goal for myself and trying to achieve it. And believe me, it is the journey to a goal you remember more than achieving the goal itself.
One of my most memorable games was when I decided to do exactly the OPPOSITE of what you're supposed to do in a civ game - I thought it would be cool to try to survive as long as I could as a backward, primitive civilization.
I chose the Aztecs, and upon entering the game, turned my research down to zero. My goal was to keep my people as happy as possible and just survive, so I cranked luxuries up to about 40%, expanded rapidly, built temples and granaries in my cities, and then built up a large military of archers and jaguar warriors to defend my borders. I had a about 11 cities, half of which were in prime locations. My northern borders were heavy with hills and mountains, so I had a few fortress cities there to maintain my borders, with roving patrols to keep an eye on my neighbors to the north and south (Rome and America respectively).
In Civ III you couldn't stop research completely, so I was still researching, just at a ridiculously slow rate (something like 40 turns per tech). I was the envy of the world in culture and my treasury was overflowing (I was so backward in tech that I had built all cultural improvements I could, then just built wealth). My population was massive, but while my enemies were fielding knights, pikemen and medieval infantry, I still had archers and jaguar warriors valiantly defending my ancestral lands. Then the trouble began.
The Romans started wandering into my lands, first a few units, then a deluge of legions. They were still in the northern mountains, where my hopelessly outclassed warriors might be able to survive a guerilla war. I had a hard decision to make - do I preempt their strike to slow their advance to my core cities so I can field more units and wear them down with attrition, or do I simply wait it out and see what they're doing? Erring on the side of caution, I waited, and watched. The Romans marched through my lands, and though I was affronted at their brazen inconsiderate trekking through my people's lands, I wanted to see what they were up to - perhaps, I thought, they were making a move on the Americans.
I turned out to be right. Breathing a sigh of relief that I had dodged the bullet of a near hopeless war, I watched as the Roman legions bore down on the American lands in the south. The war was vicious, and more than half the battles were fought in MY lands as the Romans marched troops south and the Americans marched troops north to meet them. Neither side could gain an advantage, and eventually a peace was signed, while I sat back marveling at my luck and intuition. I never got a chance to finish that game, as my harddrive died a few days later, but I will always remember it for the fun factor and the unique situation I was presented with.