The genesis of this game was Krill's request for a One City Challenge in the Realms Beyond thread. I thought about it, and connected it with another concept I'd been considering.
That concept was to use Byzantium's unique ability for something that no other civ can do, combine together two religious beliefs of the same type. These are the enhancer beliefs Religious Texts for stronger religious pressure and Itinerant Preachers for increased pressure range. They synergize extremely strongly, multiplying together effective modifiers of 168% and 165% for a product of 277% of base pressure. (This isn't a new idea, it's well known as the most exploitive use of Byzantium.)
So where do we go with that, how do we make that exciting? Domestic religious spread is easy enough anyway, so we need something with a goal of evangelizing the world. The founder beliefs are the category that operates from foreign religious spread. Hey, how about World Church? I could see that getting up to 40 culture or so, would take 200 foreign followers, which sounds doable.
And of course the game type that can exploit that is OCC, where constant culture means the most as it's a larger percentage of the needed culture for each policy and the culture victory. So there we go with Krill's request.
Here's the setup. In Gods & Kings, the strongest terrain for an OCC (or any capital really) is a swath of desert hills enhanced by the Petra wonder. After some playing around in the worldbuilder SDK, I figured out this setup as the best way to reliably generate lots of desert. Inland Sea map because it creates a lot of rivers and otherwise chunky terrain. Young world age (more hills), arid, hot.
And with that setup I got this on just the second map roll! Looks pretty nice with a northwest move. Only 7 good Petrable tiles, but everything else is the complete package: expansive riverland with floodplains, four luxuries for selling, marble for wonders, and a mountain for the observatory. Flood plains are also good; even though they don't receive the Petra bonus, they do work for the Desert Folklore pantheon which could rack up 20 faith per turn for this city.
Ruins: Archery, culture (turn 9 Tradition opener!), survivors to size 3, survivors to size 4, spearman upgrade, 60 gold, late Bronze Working!, late culture (T43 for Aristocracy.)
I bought a worker as soon as possible (about turn 12) by taking an AI loan, and another on turn 24 after selling my first copper hookup plus another loan. Build orders were my most common opening: scout - scout - granary - shrine - archer. Slipped in the watermill before starting on wonders, because both marble and Aristocracy would arrive shortly but not yet.
Here's a turn 50 overview as we start the Great Library. That 16 turns would come down with the marble quarried and second copper mined. (I'd bought the marble tile, which is virtually always correct for a luxury in third ring, the culture expansion takes too long and selling it will more than make back the buy cost.) The GL completed on turn 62, shooting Currency, and starting right into Petra.
My Pantheon had come on turn 49, which was too late: Desert Folklore had already gone. Sigh... I decided to play it out anyway. I've been start scumming a bit too much lately, let's play this one as it lies, this game isn't trying to set any records so doesn't need to go perfectly, and the desert start was good but not crazy big. We can get by without Desert Folklore. I took good old Fertility Rites, can't ever go wrong with that.
Incidentally, the mechanic of escalating pantheon costs rubs me the wrong way. It's designed backwards. First the emergent behavior of what should happen was defined (the pantheons get founded over time spaced a few turns apart) then the mechanical rules set up to produce that. That is not how an open sandbox game like Civilization should work. The macro behavior should be the sum of the micro actions. Don't put artificial bends on the micro mechanics to force behavior at the macro level.
In fact, this sums up a lot of Civ 5's mechanics and its casual appeal. Consider the escalating growth and building and policy costs. The planned macro behavior was for those to come at a mostly constant rate, so the costs and production sources (many linear additions, few multipliers) were defined to induce that outcome. Reviewers and casual players love that; they're used to linearity in other genres and appreciate the security blanket that they'll get a well-defined experience. Hardcore strategists hate that; we want to sink or swim on the aggregation of our own strategic input.
The G&K religion system follows all those same misguided rules. The prescribed macro behavior is for most but not all civs to get their own religion, produce missionaries and prophets at a linear rate, and engage in spreading skirmishes with them. And that always happens because the design is straitjacketed into that. Civ 4 throws it wide open: if you want to hoard a seven-headed holy hydra, or ignore the subsystem entirely with Free Religion, or manipulate it into a religious victory, go right ahead, Civ 4 won't get in your way. Whether you can actually get what you want depends entirely on the sum of your micro decisions against your opponents', and that is the core of what makes a strategy game.
Anyway, back to the game. Of course an OCC must go for culture victory. I believe I know the optimal social policy path, which is constant and doesn't vary with the civ or map. Tradition - Piety - fill up Patronage until industrial unlocked - Freedom - Liberty - finish Patronage. Late Liberty is for the Golden Age and finishing Artist, which produce more culture than anything in Commerce or Honor. No other options help (Commerce does very little with no roads or anger problems) and no other order makes sense.
I acquired two religious city-states by happenstance. One had been impressed by the Great Library plus a barb camp clear. And on the other side of the map, I actually knocked out a barb camp with just my two scouts. Between the CS and grabbing Stonehenge after it hung around late, that would add up to enough faith, so I was content to play out this game without Desert Folklore.
The Great Prophet arrived on turn 91. I was second to religion but didn't miss anything important (except the religion name I wanted! Christianity.) Divine Inspiration should serve as a sufficient replacement for Desert Folklore. It made the next Prophet follow shortly on turn 104, which beat Spain's enhancement by just one turn! So here's the whole religion.
After Currency, the research priority was Theology, reached on turn 92. I would need everything here. Great Mosque before Hagia Sophia is the better order, since faith would accumulate to 300 first, then HS would spawn the 500-faith prophet. (I expected said 500-faith prophet to also get the extra spread from the Great Mosque, but it didn't. Hmpf.)
I was one-more-turning like CRAZY to see if I'd get all this, the religion and all the Theology wonders. I did, then went to bed sometime past 3 AM and zombied my way through work the next day. 
Here's a look at my city at the traditional date of 1 AD. Constantinople was churning out wonders constantly, of course. Stonehenge (T83) Oracle (T91) Great Mosque (T101) Hagia Sophia (T109) Terracotta Army (T119) Alhambra (T136) Machu Picchu (T140). I was trying to have patience on growth and let it work the hammer tiles. (What Can Wait Must Wait, and growth can, wonders can't.) Machu Picchu was mostly to impress three city-states. And check out that 50 faith/turn number - that's a missionary every four turns. More in a bit.
After Theology, research up to good old Civil Service of course. Bought a third religious CS, and also bought my market and cathedral. I ended up missing one wonder, Chichen Itza, but that was tolerable. Then on to Education and a university, hired the scientists right away, needed to keep up on tech.
Hagia Sophia was a very critical piece of my plan. This Great Prophet had a very important and specific mission: Snuff out every other religion in its cradle!
Like that. Use a Prophet spread in a foreign holy city before the religion spreads, and you all but wipe that religion off the map. Barcelona and Seville would convert quickly, only needing two followers. All these cities in turn would apply enough pressure back to Madrid to outweigh the special holy city internal pressure.
So Spain was now nearly locked out of their religion: any bought missionaries would be Buddhist not Christian, and I don't think the AI uses inquisitors. Perfect. Let the World Church reign and rake in culture for my Constantinople.
A difficulty is that Prophets are difficult to navigate around the countryside. WhyTF exactly can't noncombat units move through rival units? My solution was to team him up with my original exploring scouts. With their greater mobility (both the terrain ability and moving through other units), they would form a conga line ahead of the Prophet to hold reservations on the tiles for him. This was effective, but exceedingly stupid that the movement mechanics force you into such idiocy.
Anyway, the Prophet took up residence in this spot with his escorts, between the next two civs that were likely to found religions (based on when they got their pantheons.) Montezuma dared to establish some blasphemy, and the Prophet swooped right in to wipe that out. He did the same for the Siamese and English religions too, successfully squelching every other religion in its cradle. Go go World Church. The hilarious part is that after each of these, the AI leader confronts you about "stop proselytizing", and I could truthfully say yes!
Besides the Prophet, I was also dispatching a torrent of missionaries. 50 faith per turn means a missionary every four turns. With a wider civ, that slows down as you enter new eras and the faith cost increases. But this single city was researching slowly, didn't reach the Renaissance until turn 150 800 AD, so this faith pushed out about ten missionaries in all before then. Naturally they converted lots and lots of cities -- a single spread is enough to convert any atheist city up to size 9. Smaller cities were left to convert on their own which usually happened quickly.
But dear Buddha, those traffic jams are annoying.
Here's a world overview in 1000 AD, in two parts because Civ 5 is hobbled and doesn't let you zoom out far enough for a proper overview.
Note that EVERY SINGLE CITY IN THE WORLD IS BUDDHIST. 43 culture/turn from World Church! And amusingly, those English cities were receiving pressure directly across the inland sea! They were within 13 tiles of Constantinople and the Spanish cities. That totally makes no sense, the travel distance without Astronomy is more like 25 hexes - and in fact I don't even have Optics to embark and the land distance is 33 tiles. Hah.
Hmm, three Siamese cities actually broke out of my religious lockdown. Must have been a Great Prophet. I wasn't sure whether a GP would still be for the domestic religion if a foreign religion took over the city, and if the AI would know that and actually do it. But I still had an enormous pressure advantage. Sukhothai would resist reconversion thanks to the holy city pressure, but the other two eventually converted back to Buddhism on their own.
Anyway, presently the focus moved away from religion. World Church would continue to provide a steady stream of culture, but now on autopilot and it wasn't worth chasing down the odd city or two that got prophetted back. Since I didn't need faith for missionaries any longer, I took the 800 faith Great Prophet and settled him. That was mostly to clear that threshold out of the way and let the counter accumulate higher, in anticipation of buying the 1000 faith Great Artist immediately on hitting Freedom.
But of course, research is the Achilles' heel of the OCC. I hit Acoustics for the Renaissance disconcertingly late, turn 150. (Napoleon got that on turn 117.) I had two research agreements, but in G&K they are based on your beaker production, which for an OCC is not much. Each RA went for less than 300 beakers. I made up a bit of time with a spy steal of Astronomy, but two other attempts saw the spy get killed. Later I got a few more steals - Steel, Gunpowder, and a nice prize in Industrialization, but none of those were on a critical path.
I'd been running max Artist specialists and settling the Great ones, of course. The Great Scientist counter was also accumulating, and I decided I had to let that spawn, both for the GSci itself and because I needed to continue using the scientist slots for the beakers. I scheduled it for the 400-point GArtist and GSci to spawn together. The GSci settled - the bulb was awkwardly timed and only about 800 beakers (I tested and reloaded - things like this are why I don't play for the CFC HOF), while settling would get up over 40/turn and pay back soon enough.
But to enter the Industrial Age, I had to pull out the biggest gun, Oxford University. That went for Archaeology on about turn 180. The Louvre and Pisa Tower followed for three extra Great Artists. Running the math showed that settling them rather than Golden Aging was correct: the GA would be 90 culture x 10 turns, and settling 27 culture x about 60 turns until game end. Looks like the big breakpoint for a Golden Age is the Freedom finisher, for both the GA lengthener and doubled landmarks.
Taj Mahal conveniently completed right after the Freedom finisher. I chained Golden Ages from there to the end of the game with Representation and one burned Great Artist.
Umm... City-states via Educated Elite can provide the special UU Great General? 
!
Here's a late look at the city; of course both OCC and culture get pretty monotonous late game and aren't worth any reporting detail. At size 30, I gave up on further growing the city and just worked hammers and specialists instead. (Size 30 is on the small side for OCC, one major problem being I never got WLTKD, never satisfied that Gems demand. Rammy had five but somehow never had a surplus one for the longest time, then he turned hostile.)
Built Porcelain Tower, mostly because several CS wanted it, but bulb towards Radio was helpful too. 2 more RAs matured, making six total on the game. I took artists at 1000, 1500, and 2500 faith; and wouldn't reach 4000 for the next so took one more holy site prophet instead. Added Kremlin just for culture. Never reached Plastics for Cristo.
For the first time, I made the effort to set up an overflow chain to cheat towards Utopia, like this.
Walls (any cheap item will do)
Knight 119/120 then finish it on turn X-4
Windmill 240/250 finish on turn X-3
Stock Exchange 358/360 finish on turn X-2
Big Ben 749/750 finish on turn X-1
This works to save up several turns worth of overflow to apply towards Utopia Project (or any wonder) instantly. That stepladder is needed because overflow is capped to the cost of the item, but by scaling up to a big item last, the city could overflow up to 750 hammers from Big Ben. An optimal overflow chain is when each step is closer to the next than the amount of your base production, although here I didn't have anything to fit between the Stock Exchange and Big Ben. This technique also worked in Civ 4, although not as well since there usually weren't buildings and wonders hanging around in the 300-1000 cost range.
So the Utopia Project on its first turn received 151 base + 534 overflow = 685 hammers. So that cut off three turns from the finish date. That's worth doing, if you can stomach yet another gamey misfiring mechanic from Civ 5.
Culture Victory on turn 263 1765 AD. My fastest yet in G&K, although I haven't tried yet for a truly speedy one. (That 919 culture is after a Golden Age just ended, it was really more like 1080, which is a ton for a single city.) All those units came from militaristic city-states.
+72 culture from religion - wow. Nearly double my initial projection of 40 from World Church. And that can go quite a bit higher with a better setup. Do it on a bigger map and on Emperor difficulty for the AIs to have more cities, and I think this could break 100 culture.
What I discovered in this game is how religion is ALL about virgin territory spreading. It's literally an order of magnitude easier to spread religion into cities that haven't had any yet. A missionary can convert atheist cities 1:3 with Great Mosque. But it takes like 3:1 missionaries to convert a single otherfaith city. If you want to make a founder belief work widely, speed speed speed is the key, convert the entire world before any other religions get started. And when they do, kill them at founding with a Great Prophet.
Honestly, Byzantium wasn't even needed to make this work. It could easily be done with just one of the enhancer beliefs, Itinerant Preachers the better choice because it is full strength right away and doesn't need Printing Press. I may do this again with a civ better suited for OCC.
Full build order, just in case anyone is curious: Scout (T6) scout (T12) granary (T27) shrine (T33) archer (T39) watermill (T46) warrior (T50) Great Library (T62) Petra (T77) [bought market] Stonehenge (T83) Oracle (T91) Great Mosque (T101) Hagia Sophia (T109) National College (T113) Terracotta Army (T119) missed Chichen Itza (T123) one cataphract (T125) Machu Picchu (paused T128) Alhambra (T136) resume Machu (T140) university (T145) garden (T148) National Epic (T152) [bought Opera House T152] Sistine Chapel (T166) bank (T171) Hermitage (T175) Observatory (T180) Oxford (T184) [bought museum] Louvre (paused T188) Workshop (T191) Ironworks (T194) resume Louvre (T203) Leaning Tower (T210) Taj Mahal (T218) National Treasury (T220) [bought Hydro Plant] Porcelain Tower (T226) Constabulary (T227) Public School (T231) [bought Broadcast Tower] Factory (T234) The Kremlin (T239) Grand Temple (T240) and then it was setting up the Utopia prebuild overflow chain.