Adding to
Fantabaldu's Excellent List...
The DeLuca Family - Mafia Game totally focused on story and setting a high bar for quality; you are roped into servitude to a Family by a mysterious contract and must survive and rise through the ranks
Happenstance - a Sci-Fi setting where you, as a very poor Powered Person are under attack from an unknown enemy; more than this though, the storytelling will make you love your girlfriend (crazy, right?)
MidLife Crisis - This one is just fun, somewhat ridiculous, but absolutely awesome. The title says it all: you're a man in a midlife crisis and boy oh boy do things go off the rails (in the best way)
Light of my Life - If you can tolerate the art style, you will love your daughters long before you reach the end of content. Some people refuse to get past the art style though, which is simply their loss.
Being a DIK - A top-tier college game with stunning characters and an incredible story. Go play immediately.
Depraved Awakening - A classic and must-have addition to any list of this sort. You're a detective solving a crazy case; intrigue and murder ensue
Hahaha I see you added The DeLuca Family while I was compiling this,
Fantabaldu ... Well, it deserves a double mention so I'm leaving it