Interesting thread. I played through about ten cultural games on deity during last few months, since domination victories were getting a little repetitive. The culmination was a t.232 win with 4 city India start and luckily placed Constantinople next door, stocked with temple of artemis and the leaning tower of pisa (double jackpot!), which i conquered, was certainly a deciding factor.
Anyway, i tend to think the fastest (outside sacred sites) cultural path lies through a little bit of warring with crossbows (or frigates, if the map commands it). Liberty is also a viable option, like Manannan suggests (especially when we compare it to honor), however tradition doesn't restrict gameplay to four cities, creating a synergy between peaceful initial expansion and mid game (t.120-140) annexation of a couple strong cities. Sure, annexing 10 more cities is also a way to go, and certainly a fun one, but i find this way to be generally slower than healthy "hit once or twice and consolidate".
Interestingly, i tend to use the same tech. path when going completely peaceful and when doing some warring mid-game. It goes like this:
1. Acoustics. Sistine Chapel helps a lot when trying to fill 4 Rationalism policies fast. I do pick aesthetics opener first, obviously. Sometimes, when i'm able to get the Oracle, i get two in Aesthetics, before going deep into rationalism.
2. Guilds, Machinery, Printing Press. I save 660 for the university in the capital. The rest of the cash is for upgrading composites. Leaning tower is very tempting. Always. But i slap myself on the wrist and get 4 knights instead. Sometimes i give in.
3. Architecture. Mainly on the way to scientific theory, but, occasionally snagging Ufizzi or early Hermitage can add to the confidence.
4. World's fair. Usually hits at the most inappropriate time - when i'm hard building public schools. But, pausing the schools is usually worth it. The only viable workaround here is to finance purchase of public schools through spoils of war. Very strong move, actually. I always "waste" one writer few turns after fair is won. Yields about 2k culture, on a good day.
5. Ideologies. Opening Radio with Oxford seems to be a must. Eiffel tower.. bought with faith, usually. And, of course there is little reason to delay building archaeologists like crazy from that point onwards, which is around t.180, on average. Squeeze the factory in here, rush a museum in there...
6. The final push - fertiliser, research labs, internet, then airports, if needed.
7. I try to place writer's guild somewhere outside Cap. Running 2 guilds and a Uni in the same city is.. a little too much for me.
Generally, t.240 - 270 CV's are likely with this path. The range is so great as there is little way to predict the depth of cultural wells of your opponents. However, army is strong ally, which is able to shift odds and speed up wins, where as peaceful approach is a little bit of a gamble.
