What do you most enjoy in a game?

The games I enjoy best are the ones where I must make "agonizing decisions" based on trying to guess what my opponents are going to do. "If he does THAT then I'm okay if I do THIS... unless Fred does this OTHER thing which will destabilize that standoff over THERE that I need to keep stable, in which case..."

I need lots of decisions that will each have a significant affect on the outcome of the game; I need to face so many opportunities and/or threats that I can't summon the resources to deal with them all easily; and it all needs to depend on what the other players do.

Some wargames fit this model: A Game of Thrones is probably my very favorite game, when I can get four other people together to play it with me. War of the Ring is another example. But games like Tigris and Euphrates and Through the Desert also fit the description.

I like games to have a mix of luck and strategy, but light on luck and heavier on strategy. If there's too much luck involved, my decisions don't mean much and that removes the all-important agony. Of my example games, AGoT has only a small random element, and all battles are deterministic. WotR and T&E emphasize strategy over luck, although T&E has rather more luck than I might prefer. TtD is pure strategy and has no luck at all.

(Just because this point comes up from time to time: by "luck" I mean deliberately random factors. My opponent's behavior may sometimes be unpredictable to me but it's not random. Dice and shuffled decks and tiles blind-grabbed from bags are random.)

I like themed games, but I don't require that the theme run deep. The camels and palm trees in TtD are fun to play with even though the theme is shallow and the game could be completely abstract. But I especially enjoy immersive games, where you just disappear into the game world for hours on end; this seems to require deep theming. For me, AGoT and WotR are like that. Role-playing games are like that too (or else they're not worth playing, IMHO!) although it's been years since I've actually done any role-playing.

There are no particular mechanics that attract or repel me. Auctions, dice, building, battles, tile-laying, route planning, they're all good. It's the playing experience that matters to me.

Altered State:

That is what I most enjoy while playing a game. If I can be transported from the kitchen table to the boardroom of a real estate tycoon, then I am happy.
When I am playing Axis and Allies, I am the General!
Abstract games provide a different sort of transcendence - athletes call it the flow - and no game provides that rush like speed chess. I agree with Foamy.
Incidentally, I get the same feeling whether playing chess online or OTB.

Mitch

I like the following things in games (and try to place them in my creations as well):

1. When you don't know who'll win until the end. It shouldn't be a complete surprise everytime, but it helps keeping the players at the edge of their seats, eagerly waiting for their turns.

2. Actually, quite the same as one. Players should be able to participate throughout the whole game, straight up to the end. (I'm working on a secret-character game and it was hard, but I've managed to have all the players involved the whole time).

3. When you have to pay attention to what another player does in his turn. To be the fastest with a trading offer, of because he's giving away information etc.

4. Once you've earned points, they should be yours. No one should be able to steal them.

5. The theme and the game should be one. (By using a good and original storyline)

6. Not too much luck. I'm never that lucky.

Well, that's about it for me.