Somehow though I get the feeling that Mission's interactions with Outpost and Possession are going to be nightmare-inducing.
I don't think it'll be too bad.
First off, Mission does not interact with Possession, since you can't play Possession on Mission. Possession will only allow you to take one extra turn with Outpost because Outpost by itself can't be used to take more than two turns in a row.
Oh, wait,
Possession, not
Precession.
Let's deal with the simple case of Mission and Outpost:
If you play Outpost, then buy a mission, then in your discard phase you will keep out the Outpost and then (probably) get to choose which extra turn you get: the mission or the outpost extra turn (similar to how playing two possessions on an opponent, and playing an outpost during the first posession allows the possessor to choose to play out outpost extra turn before the second possession, or after). Playing the mission first results in a mission with only three cards, and no third turn. Playing the outpost extra turn first results in a third turn in a row, the last turn a mission.
Now for Possession:
Cribbing from the Wiki on Possession/Outpost,
Interactions with Possession are difficult: remember that when your opponent Possesses you, he makes you take an extra turn, and makes all decisions for you.
- So, if your opponent plays a Possession, then makes you buy a Mission during that Possessed turn, they will get to see the five cards you draw; then you will take your buy-less Mission extra turn, since that will only be your second turn in a row; and then you will take your regular turn.
- If your opponent plays two Possessions, you take two extra turns from them; so if your opponent then makes you buy a Mission on the second Possessed turn, it does nothing since the previous turn was yours.
- If your opponent plays two (or more) Possessions and makes you buy a mission during the first Possessed turn, they will get to make you decide when to take the extra turn from Mission; they can make you take an extra turn with no buys cards after the first Possessed turn, or they can choose to delay your Mission turn. Unlike the Outpost case, you will get to play the mission, since nothing in the Mission card says you can't have more than two turns in a row, just that you can't get them from Missions after your first turn.
I would assume that if my opponent plays two (or more) Possessions and makes me play both an outpost and buys a mission on the first Possessed turn, they would get to choose if my second turn was the Possessed turn, the Outpost turn, or the Mission turn.
What is unclear to me, in both the double-Possession/Outpost and double-Possession/Mission cases is who controls on the Outpost or Mission extra turns? Would it be I'm Possessed, I control the mission, I'm Possessed, I control my normal turn?