Tweet
I wasn't going to bother posting, but KarlMarx9001 said it should be virtually impossible in a peaceful manner, so here goes 
I did it last Wednesday 12/5 pre-patch. I did a tiny board with high water and archipelagoes. Virtually no war. Cultural victory. I won with something like 70 turns to spare, if I remember correctly. Basically, you must have the Colossus, Great Library and Shakespeare. I got three others as well, but I can't remember which ones off hand. Without them, you don't have a chance for a cultural victory. The Colossus brings extra money, the Great Library catches you up on tech in the early game, and you NEED the happiness from Shakespeare. I was pretty much 5 techs or so behind the whole game, but I didn't really care since I knew I have the culture to win. I think I went to war only three times, and they were all defensive. I have the Romans and English to my North, and the Egyptians to my south. I never even got offered a cultural takeover a city because the two closest cities to mine were capitals! It was an odd game, because it was slow going tech wise, but I think it was because the Romans kept going to war with the other two (eventually wiping out the English, and hamstringing the Egyptians) and there was little tech swapping. I had a MPP with the Romans after he kicked the other civs off our large island for about the last 60 or so turns of the game. Anyway, I wasn't sure if someone had done it already, so that's why I didn't post before.
I enclosed the save game from a turn before victory in case anyone wanted to check it out.
I did it last Wednesday 12/5 pre-patch. I did a tiny board with high water and archipelagoes. Virtually no war. Cultural victory. I won with something like 70 turns to spare, if I remember correctly. Basically, you must have the Colossus, Great Library and Shakespeare. I got three others as well, but I can't remember which ones off hand. Without them, you don't have a chance for a cultural victory. The Colossus brings extra money, the Great Library catches you up on tech in the early game, and you NEED the happiness from Shakespeare. I was pretty much 5 techs or so behind the whole game, but I didn't really care since I knew I have the culture to win. I think I went to war only three times, and they were all defensive. I have the Romans and English to my North, and the Egyptians to my south. I never even got offered a cultural takeover a city because the two closest cities to mine were capitals! It was an odd game, because it was slow going tech wise, but I think it was because the Romans kept going to war with the other two (eventually wiping out the English, and hamstringing the Egyptians) and there was little tech swapping. I had a MPP with the Romans after he kicked the other civs off our large island for about the last 60 or so turns of the game. Anyway, I wasn't sure if someone had done it already, so that's why I didn't post before.
I enclosed the save game from a turn before victory in case anyone wanted to check it out.