So we've now established that a session is the frame in which a game needs to develop its ideas and create impact, and we know how this session is defined: logical stopping points. Bookending save points, quests, or a flexibly sized series of micro-tasks. We acknowledge that it's important for every session to have a beginning, middle, and end, not just every quest/level/task, because once the session ends, the game is out of the player's mind. Now the question remains: if each sitting is its own unique experience, how do we justify that a game can be composed of anywhere from one to a hundred of these? How do game sessions differentiate themselves from repeated viewings of the same movie?
(by the way, I notice at this point that some kind of obvious comparison to television exists, because these "sessions" I'm discussing are awfully similar to "episodes". The biggest difference is that games (like books) are terminal, while television programs often aren't. TV can meander and float about without any overarching purpose. Regardless, I find it no more sufficient to end the conversation by saying "TV works, hence video games work" than I do to use film or literature identically.)
Fun is a factor, but saying "make the game fun" makes me feel a bit brain-matter-deprived. If you can't trust your game to be good enough that players can't put it down, give them something to come back for. Dangle the carrot. Good example: XCOM. It's fun sure, but is that really what makes it addictive? The game keeps its portions small and the rewards coming, so you can stop anytime you want. It makes sure you don't want to stop (and that you want to come back after you do) by tying rewards into a linear chain. As soon as you reach one, the next becomes available. You play in anticipation of the immediate goal, then as soon as you reach that and start enjoying the payoff (like trying out that new lazer rifle), a new reward pops up on the horizon. This can be dragged out in the way that MMOs, action RPGs, and any games revolving around the word "loot" do it: randomizng the cycle. You don't know what the next treasure will be, and you don't know exactly when it'll come, but you know there's a virtually infinite supply out there and that it'll show up eventually. This establishes a player mentality of "maybe it's around the next corner" and keeps them coming back indefinitely, but gives enough reward that they can stop anytime and still be satisfied. Dark Souls uses an even more direct dangling carrot: place the reward right in front of the players face, just an inch out of reach, and then make it impossibly difficult to take a step forward.
It's not just a betimes dubious reward cycle that keeps us coming back though. That's implying a primitive dedication, an almost involuntary urge. Those games can be fun, challenging, and satisfying, and make for a great way to spend time, but let's not forget that ultimately we're discussing a narrative medium. Forget indefinitely repeating action for a second and look at how to promise a return will be worth the time. This seems an almost trivial question; make this session good, and they'll assume the next one will be. But each subsequent session requires a little bit more to make it play-worthy. What was awesome once isn't necessarily awesome twice, and you can't exactly get away with using the same plot point twice. It seems idiotic to even need to say this, but the entire length of a game needs to be supported with original content. A designer needs to be able to judge how many times they can repeat the same sequence before it gets old, and how draw that out by interspersing different ideas. How long can you keep me in one village, and how long after I leave it will I want to go back?
This structure is open for escalation and variation, and allows one game to tell a lot of different stories, as a book or TV show may. I'll get more in depth with this some other time, but for now it's the only ending you're going to get for our discussion of session play.