Let's say I've somehow managed to get two or more Exquisite Archangels out onto the battlefield:
Something causes me to lose. Let's say it's just regular creature damage, and my life will reset just fine with no shenanigans.
What happens? Do all Archangels get exiled at once, or do I exile just one and have one more chance left for each extra Archangel beyond that one?
Only 1 Archangel gets exiled each time you lose. This is because the Archangel has a replacement effect, and when you have multiple replacement effects attempting to modify the same event you do it one at a time. The first Archangel replaces you losing with her effect. You are now no longer losing so the remaining Archangels don't have anything to replace so their effects do nothing.
616.1. If two or more replacement and/or prevention effects are attempting to modify the way an event affects an object or player, the affected object’s controller (or its owner if it has no controller) or the affected player chooses one to apply, following the steps listed below. If two or more players have to make these choices at the same time, choices are made in APNAP order (see rule 101.4).
616.1a-c These rules don't apply to this case.
616.1d Any of the applicable replacement and/or prevention effects may be chosen.
616.1e Once the chosen effect has been applied, this process is repeated (taking into account only replacement or prevention effects that would now be applicable) until there are no more left to apply.
You choose one to apply, and it saves you, and the other one sticks around until the next time you would lose.
This is explicitly stated in one of the card's rulings in Gatherer:
If you control two Exquisite Archangels, you choose which one’s effect applies. The other’s effect won’t be applicable after that until the next time you would lose the game.
You will exile just one of them.
When you would lose the game, both Archangels try to replace "lose the game" with "exile this and reset your life total". Since multiple replacement effects are trying to apply at the same time, you choose which one applies first. Regardless of which Angel you choose to use, once you've made your choice, the other Angel will no longer apply, because you're no longer about to lose the game. You'll exile the one angel, reset your life total, and then continue on your merry way.
Also, as was stated in another answer, this is explicitly called out in a Gatherer ruling as well:
If you control two Exquisite Archangels, you choose which one’s effect applies. The other’s effect won’t be applicable after that until the next time you would lose the game.
