Hmm, let's see...
Originally posted by Hat KidFire Emblem Genealogy of the Holy War - Peak FE. Jk, but it is at the top for me. The gameplay nowadays can be a little annoying (no trading, obtuse doubling mechanics) but the story, graphics, and music are awesome. The generation system is really interesting and allows for a lot of experimentation and potential baddening of your 2nd Gen units, intentional or otherwise. Plus, it feels natural, unlike the child mechanics of Awakening and Fates (especially the latter) which felt shoehorned.
First of all, ^ this nearly verbatim. Personally
Shadows of Valentia is my favorite FE, in large part for it's presentation, music, voice acting, and how all of those are used to turn a now nearly 30-yo NES game into a story that never fails to invoke feelings in me (Not to say it's without it's flaws, of course). Genealogy is a close second, though, both for the reasons Hat Kid stated and due to the way the in-game maps directly correlate with the world map, and how the game uses those to draw a political narrative.
As for other franchises:
Pokemon Emerald - might just be childhood nostalgia, but something about this one just always holds a special place in my heart.
Octopath Traveler - Funnily enough, I only purchased this because I got hooked on a few songs from the game. Of course, I learned to love much more than just the music from it, and even though I'm probably only halfway through the game after 60 hours of playtime, it's been a beautiful experience thus far.
Minesweeper - Does this even count? Eh, why not. I grew up with it and only got back into it recently, but it's a really good demonstration of performing logical proofs on isolated instances and using those to more quickly and efficiently solve larger scale problems (or maybe my programming brain read too hard into it, haha).