Crusader Kings II is, well, kinda emergent I guess? Your objective is (IIRC) to get as much score as possible - Prestige, wealth, and Piety, and you can set mini-goals for yourself through the "ambition" system.
Aside from that, however, anything goes. By hook or by crook is pretty much how things seem to be done.
Do be warned that a good deal of the game's content is DLC.
Also, a lot of weird shit can be found in CKII - check out r/shitcrusaderkings2says or something like that. Maybe it was just r/shitcrusaderkingssays. The "learning scenario" directs you to plot your own brother's death - it was the Middle Ages, things were kinda like that back then.
Darkest Dungeon has, with the exception of one gamemode explicitly stated to be the harder of the three available difficulties, no time pressure to complete the main quest at all.
Your heroes are all randomly generated, as are the quest maps. Unpredictable combat (maybe you'll finally hit that backrow asshole, or maybe he'll survive long enough to land three crits one after the other) and relatively high-stakes (the amount of time and resources spent on getting a hero to survive to Resolve Level 6 means it hurts on both a practical and an emotional level when they die).
There's some DLC content in the form of a new area (The Crimson Court), an endless mode (The Color of Madness), and new heroes (The Crimson Court, Shieldbreaker, Musketeer [Sort of]).
Heat Signature is basically scifi sneakybeaky Hotline Miami. Each character you play as has their own personal mission - something bad has happened and they need to do the mission to make things right.
But before they can perform the mission, they gotta get some money first, and that requires wetwork. "Kill this person", "hijack this ship", "steal this thing", and so on.
The random generation of levels, guards, and objectives means that your tales may end in triumph, defeat, or even just somewhere in between.
Mount & Blade Warband is a game where you don't really have any specific goal. Once you have enough experience, money, land, and other possessions like that, you can retire your adventurer to get a final score listing.
Between the start and the end, though, a lot of things can happen, but a lot of the time they involve building up a small army so you can eventually bring big armies to bigger battles and have a hand in the fates of kingdoms.
I might as well describe some of the other games (though I might have exaggerated when I said over ten including ones I don't play, since some of the "storytelling" there requires a good deal of imagination on the player's part).
Oolite is Elite, but free. Like Mount and Blade: Warband, you're thrust into a world with the bare minimum of necessities and you need to scratch out a living. Rather than a sword-fighting game, however, this is more akin to a flight simulator of some kind, as you spend a lot of the game in the cockpit of whatever ship you own.
I must warn you that travelling in between planets can get a little... tedious.
Dwarf Fortress's emergent storytelling is probably pretty obvoius - The sheer number of stories that have been recorded will probably do the arguing for me better than I can.
Cataclysm: Dark Days Ahead is, well... a lot of changes are happening to the game, but the random generation of cities (and therefore item placement) mean that, again, your survivor's story may take all sorts of different twists and turns.
The old XCOM games, like the new ones, have soldiers you build up to a degree. Like in Darkest Dungeon, losing experienced soldiers is both surprisingly easy and surprisingly crushing - though admittedy somewhat less so since in old XCOM you have explosives, not affected by firing accuracy.
Streets of Rogue - Granted, I only own the demo, but I still think that it qualifies, since, again, quest for completing missions can go all sorts of ways.
Survival Crisis Z - Doesn't run too well on modern OSes, but I think it's worth a mention since it's free. Isometric arcade-ish survival game about trying to stay alive in a city filled with the undead.
Aurora 4x - I don't understand this game at all, but if it's like Dwarf Fortress at all, there's probably some stories that'll unfold over the course of gameplay.
I dunno if any of them are what you're looking for, but in any case, that's the list.