With a coastal cap (particularly on map types where there is a lot of coast that the AI will want to settle quick), then some leaders (mainly HC and Izzy) make a good shot for running religion even on immort (and once I've managed to make this work on deity). In this case you are looking at the fact that early religious spread is driven by trade linkages and long before the AI can send off their hoards of missionaries with either mons or OR; you will get natural spread via sailing. On something like Archipeligo this works
extremely well for Izzy (on deity I founded Hinduism with her while building work boats for coastal fish I did have two forested plains hills to work, was settled on plains hill, and had no early resources to work aside from fish/clams; then teched sailing -> masonry -> mono -> BW which gave me 2 religions in the cap and the GLH. Needless to say, any city anywhere on the coast I founded paid for itself.
More commonly I will chase CoL, Theo, and Philo for their religions with Mono an oddball thrown in at times. The easiest two headed hydra to get is Confu/Christian; go place a blocking city where 3 or 4 border pops will secure vast swathes of land; if you have a choice you want lots of food. Tech order is minimal worker -> poly (my pref, if I miss the oracle this is better) -> priest -> writing. Personally, I like chopping the oracle in my expansion city so I can slingshot CoL. This almost invariably drops Confu in the oracle city; which is massive culture. Now we move on to researching worker techs -> masonry -> mono -> med -> theo (bulb). As you have only two cities, one of which is the cap, you found Christianity in the same city.
In theory you can then run scientists in the cap and use your second GP to bulb philo for a triple headed hydra. More practically, a strong start (i.e. gold/marble in the initial BFC), may let you detour to masonry/mono up front and if you can settle just prior to popping mono you can score a triple there. Quad is again possible, but you will need to REX behind your block after that. Moving up to quint requires a dedicated Priest effort; philo is huge as is Ramases. A quint hydra with a good block can make your RE insane; particularly with strong trade routes; it will take a long time to get the GPr needed to shrine them all (considering you pretty much have to burn 3 GP on bulbs to get them founded); but late game you can run 100% slider with ease.
In general this doesn't effect your diplo too much; must of the world is still going to be divided on Buddhist/Hindu/Jewish lines and you can always muck around by using your confu missionary early to make someone odd man out.
Chasing the middle religions (Confu, Christian, and Tao) is quite worthwhile for a FR based game in its own right. You cannot count on the AI to ever give you those; while odds are certain for one of the early religions and pretty good to get two from either auto-spread or missionaries. Having +5
from religion and having a cottage B cap with multiple mons and an academy is rather strong. All three of them come off strong techs which can all be used to nab lib as well. You may never run these religions (aside from a few turns to game the AP), but you still get
,
, and
from them.