Probably not the greatest or most original in terms of story, but the Gears franchise has some great world building, especially if you also factor in the books etc. For a series that didn't even exist before the 360 era, Gears has an incredible amount of lore and content.
Metal Gear Solid 1-4 is the favorite hands down! The music, the story, the characters.. It truly was a journey and I loved every game, every moment. It deeply impacted me and I was so emotional drained after every game. I so thankful for the experience and I hope to play them all again.
Then yes I actually like FFXIII, all three but especially FFXIII-2. I didn’t expect that I would like the games as much as I did, but it was a lovely journey, much thanks to the person I played with. And I really started to care for the characters especially Noel, Yeul and Caius..
Resident Evil 1-6 + code Veronica! The power level gets more and more insane with every game, but honestly I love it! It’s fun games, amazing gameplay, I like the story and I absolutely love the characters. The remakes where perfect, amazing to play and spoopy. Have some really nice memories from this games. Now I’m just waiting for the Code Veronica remake!!
FFXIV. Every expac could be its own standalone title, and the main story is definitely linked between all of it (and ongoing!). It's a fantastic story featuring complex, multilayered characters. Even if you don't like MMOs, it's worth checking out for the 1P campaign alone. It does occasionally ask you to group up for a dungeon or boss, but it's worth it. In Shadowbringers you don't even need to do that for the dungeons with the trust system, and just bring NPCs along.
This is an easy call, when comparing stories the ongoing Trails-series is by far the deepest, complex and most fascinating web of interconnected stories that I have ever played. There’s really not even a contest here.
Quest for Glory I-IV (I never did get around to V — is that even possible now?). The stories are self-contained, but carrying that same character over across games really did turn it into a hero’s journey.
Wing Commander I-III. Hero to traitor to saviour to Luke Skywalker. Three games, each at the fore-front of in-game cinematic storytelling.
I would definitely go with either the Witcher or Mass Effect series. However, one that hasn't been mentioned yet is the recent Hitman - World of Assassination Trilogy. Though the story is not as good as the other series, it does come together when you play all three, and I like how if you own all three you get them in the same game with all of the enhancements of the most recent release.