Since I think expansions are generally desirable for games, if only for variety, I’m choosing to interpret this as “games with expansions that you should never play without”, and mostly for new mechanics rather than content.
Eminent Domain - the Escalation expansion. Eminent Domain is structured such that in the base game, the only technology you will ever actually get to use are a few 3 point techs. By the time you’re able to get the bigger ones, the game will be ending imminently. Escalation doesn’t fix the pace, but it does implement a mechanic of asymmetric starting positions that let you start with those more powerful techs. And instantly, it’s far more engaging.
Eldritch Horror - the base game just doesn’t have enough stuff for the Ancient Ones you get, so you’ll be repeating experiences almost immediately. Forsaken Lore roughly doubles their various decks etc. All the Ancient Ones in that and the other expansions come at that level of variability to begin with. Arguably you also want Strange Remnants or the big box expansion that introduce the Focus action and focus tokens as a mechanic, but I don’t remember which those are and that’s not as mandatory. (Beyond those, it’s pretty much just content. I think the very last one also introduces a new action but it’s a lot less broadly useful than Focus and doesn’t really address a core issue of downtime/dice luck the way Focus does.)
Flash Point: Fire Rescue - hard agree re: Tragic Events. Transformative. (I did like it beforehand but I would never go back now.)
Shadowrun: Crossfire - in the base game you have one scenario you can run until you’ve earned like 15 karma, which is a lot of successful completion of that one scenario. Sure, much of the setup is randomized, but, blah. It’s also a bit light on content variety. The one expansion, High Caliber Ops, adds more scenarios of various challenge levels and more stuff, really pretty necessary IMO. Honestly it’s still a bit lacking though.
Honorable Mention: Darkest Night - each expansion for the first edition added seamless interlocking mechanisms you would swear were always part of the game and enhance it immeasurably over the previous version. (Except for #5, which was just assembled promos.) Why honorable mention? 2nd edition comes with all of them out of the box.
Sortakinda: Near and Far - the Amber Mines expansion offers a coop mode. I’d much rather play it coop. YMMV. Dice Throne - Dice Throne Adventures, for similar reasons. Except baseline Dice Throne is still one of my favorite games tbh.