I tried to maximise great people output in my 2nd try and ended with legendary (50K) culture in 1590. Obviously not enough to beat patagonia's great score, but well double the culture I got in my first attempt.
I built the oracle, Pathenon and the Sistine Chapel in the capital and founded 2 religions (hinduism and Judaism) there.
In the beginning I picked CoL as my free tech from the oracle, built a temple in the capital and generated a great prophet for theology (for something to trade and the sisitine chapel) and founded a 2nd city to build military and a library (for a great scientist and philosophy (=pacifism)). After founding taoism I researched literature (for national epic), traded for mathematics and researched music for the extra great artist. Got beat to liberalism by Hatshepsut by 2 turns and stopped research there.
I founded 3 additional cities, who build military during the whole game (nearly) I was quite concerned, because I adopted Hinduism, when Pacifism was available and everybody on my continent was buddhist.
In the end I generated about 120 GP-points per turn, so I got a great artist every 10 turns (2 in the 16th century!). Not having the hermitage hurt a bit though.
Heres the save:
http://forums.civfanatics.com/uploads/39198/Challange_3-culture_AD-1600.Civ4SavedGame
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I gave it another try

This time only founding Hinduism in the capital, building a warrior, worker and 2 settlers then.
My aim was to produce as many great artists as possible, so I wanted to get into pacifism quickly, get the music great artist and to get to liberalism first for nationalism. The Hermitage obviously is a key building in this challenge, so I tried to shorten the time to liberlaism by founding two cities ASAP, one for the GS-library(philosophy), the other one for cottages (and rice)this time. Later I founded a third city, near the iron, just in case that I need military.
Everything went well, I scored the Oracle, the Parthenon, the chapel (in the capital) AND the
Hanging Gardens (in my 2nd city) and built cottages everywhere (except the capital) to be able to get to liberalism (nationalism)
quickly. I traded my marble and horses for more health resources and made it to liberalism first, picking Nationalism as my free tech and chopping my first forest next to the capital, but only one to keep my +1 health bonus from the remaining two forests. I completed the Hermitage around 1150 I think, researched drama and stopped research there.
The capital build health buildings (not monasteries

) and was working only food tiles if possible most of the game. No Notre Dame, no Taj Mahal there... -- Building a grocer there, probably was a mistake, cause it gave me +2 health for 15-20 turns, but I wasted 8 turns building it...
Biggest pop Kyoto ever had, was 21, but unfortunately I had to starve the city down to 19 to get the last great artist out in 1600 AD giving me 75545 culture with his last great work
The save :
http://forums.civfanatics.com/uploads/39198/Challange_3-culture_AD-1600-2.Civ4SavedGame