Keane (nothing to do with the rag ****!) Stars a much younger Damien Lewis. No music, all natural lighting, a really stripped back film with no special effects, about a man searching for his missing child in NYC. Subject matter not for everyone but I seem drawn to bleak films so it was right up my street.
Talking of bleak films, as mentioned previously up thread Snowtown is excellent, Kevin Bacon is superb in The Woodsman, Detachment, Wilbur wants to kill himself, Paradise Now is an excellent foreign film focusing on 2 suicide bomber friends, Blue Ruin.
Love the Coen brothers as well so most of their films are off beat to some degree. Really enjoyed Inside Llewyn Davis.
Never really been one for flashy Hollywood blockbusters, give me a gritty drama or quirky black comedy every day of the week :)
Oh and not sure they qualify but Memento and American Beauty are both genius, albeit not totally off beat.