Why is Possession hated so much? I don't have Alchemy, but it still seems like a decent card.
"Because it leads to degenerate games." This has been mentioned several times, but I wanted to show a clear explanation of why this is the case.
If player A goes for a normal strategy (say 1 province per turn), player B can simply go for a strategy of "play more than one possession per turn". Since both players are using the identical deck to score points (player A's deck), player B will basically always win because they get to take more turns with that deck.
Keeping that in mind, both players will try to build a deck that can't do anything except play possession. Which leads to a stalemate. Even when stalemates don't happen, the games tend to not be very fun because you're playing with decks that were designed to not be good.
It is even worse than that. Suppose Possession is the only way to gain an extra card each turn - no other +gains in the kingdom at all. You gain your first prov, I get Possession (swapping a Silver to a Pot).
With reliable decks you get prov 2, I get gold/prov 1. You get prov 3, I get prov 2(my deck) and prov 3 (your deck).
Now it is back to you. You can get prov 4, I go duchy/prov 4 and win.
It gets worse building out to double prov engines. I get my Possession when you get to $12. You get two prov, I get two prov/duchy.
Even when the decks are unreliable, Possession typically wins. After all, Possession can trash or discard-for-sift more as it needs only one card to gain whatever the other guy gets, worse with 2 gains the Possession player can opt to buy reliability cards (Scheme, Haven, Forum, etc.) on busted hands and still maintain score parity.
It is freakishly hard to outrun Possession when your points are coming from standard green, colony, or the current online implementation. Gardens and slogs can make a go of it because that such an easier price point to hit than $6, but there really is no card in Dominion that can be more space efficient than Possession. Sure, the odd money board can outrun it when you cannot build an engine or otherwise make Possession reliable ... it is just that Possession is literally the easiest card in the game to make reliable if your opponent doesn't degenerate.
Trash the rest of your deck? You can easily gain a province/turn. Run with playing 4 Warehouses a turn? Sure, whatever.
Back in the day it didn't seem so bad, but back in the day people had not figured out how quickly engines could get up and ruining. As money strategies have increasingly declined Possession has gotten stronger. I mean when you can legitimately run Wisp engines it has to be a pretty bad board for Possession not to be idiotically strong.
Ultimately Possession was supposed to be held back by the opportunity costs of buying a Pot, having it in your deck, buying a Possession, and carrying it in your deck ... but it just doesn't work with efficient engines as buying the Pot eats one turn, buying the Possession eats another ... and then gives it immediately back next go round. After two rounds, Possession has the whip hand. Carrying around two stop cards is a bit harsh ... but not insurmountable (particularly as you can trash the Pot back out sometimes with non-terminal trashers). And you have to be awfully ballsy to trust that your deck will win just because you won the "Lab split" 6:4.