Saints Row first game was weird in tone and didn't have a coherent world, but SR2 really pulled the brakes off and went for it. I spent a lot of time just driving around and re-trying the Activities (which became the core gameplay focus from SR3 onward), AND the story was surprising and dark.
Mafia is not an open-world game, I've repeatedly heard. Is it actually open?
Sleeping Dogs
Not only a fantastic story, but innovative gameplay shifting between guns and melee that Rockstar has struggled with in EVERYTHING except
Bully. Unsurprisingly, United Front Games developers had a number of Bully team members on staff.