This... is surprisingly difficult.
It needs to be story-driven, and the story needs to be driven through relationships. Strange as it feels to admit, I think I slightly prefer being railroaded to sandbox games, but I enjoy either as long as the story is worth following and the relationships are vivid.
Notice I don't say "the story is good". "Good" can have an implication of "nice" or "happy", and I don't necessarily want that. The story needs to be strong and powerful, without the implication of violence or physicality: a story (and game) that engages my emotions is what I'm after.
There was a scene in our Aberrant game, where my character went to see her parents only to be entirely blanked by her mother. It was incredibly powerful, a culmination of of the relationship I'd created in my back story and the way my GM had built on that. My GM apologised to me afterwards, but it was a wonderful, intent moment of roleplay and I loved it.
I don't mind the setting: give me NPC's and/or other player characters to interact with and I'm happy.
And the other player character's bit is important: the fantastic NPC's and the detailed story line are a huge part of why I love our Aberrant and Exalted games, but the other player in the group wants much the same from a game as I do, which helps enhance the experience for both of us. Between the 3, we build deep, complex stories and relationships, end up nearly in tears from time to time and have an amazing time.
And yes, it does descend into daftness from time to time, and I love that too - but I love it in the same way as the Porter's speech in MacBeth: as a comic diversion in juxtaposition against more serious themes. And anyway, as the great Late Sir Terry Pratchett said, the opposite of "funny" is not "serious"; the opposite of "funny" is "not funny".