I'm playing a game in which there is a very annoying AI player whose religion continually spreads to my cities. It's beginning to annoy me, and I want to fix it. Is there any way for me to completely eradicate a religion? I don't just mean get it out of my civilization, I want to remove it from the game.
If you manage to convert all the cities that follow the religion to another religion (your own would be ideal...) then you can effectively wipe out a religion. Prophets, missionaries, and so forth created in a city follow the dominant religion of that city, and without any cities where that religion is dominant, there will be no way to create more of these. Of course, starting with and focusing on the owner of the religion is a good way to start - they're the ones who are more than likely the biggest spreaders of the faith.
This means you're going to need missionaries and prophets, and lots of them. In your own cities, you can station inquisitors, which will prevent the prophets/missionaries of another religion from spreading their faith.
This, in turn, means that you're going to need a large amount of faith in order to produce all of these units. You may wish to consider whether it's a better idea to just produce the inquisitors, and perhaps enough missionaries/prophets to convert some portion of the world, rather than aiming for global religious conquest - there may be better ways to invest your faith.
Yes, but it's more difficult than @Agent86's answer implies.
Converting every city that follows a certain religion is not enough. Unless you wipe out the civ who founded the religion, they can use a Great Prophet to restore their religion to a city of their choice, even if their religion had no cities when the great prophet spawned. Most AI opponents are really touchy about their religion, and will gleefully throw great prophet after great prophet at your encroaching religion, if given the chance.
If you manage to wipe out the offending Civ, only then can you guarantee that the religion will not be able to come back.
Razing the holy city (assuming it's not the capital) is an option, but since Holy Cities produce 5x the normal pressure (though self-targeted), you'll need at least 5 nearby cities (within 10 squares) of your religion to keep it down (if it's already your religion), or 6 if you want to convert it (slowly) without spending a missionary / great prophet / inquisitor.