Best: 'Dead Man' (1995)
Yes, you could blame Jim Jarmusch's psychedelic Western for giving Depp the false confidence required to play Tonto (see below) so many years later. But the film is simply too damn good to find fault with in any way, shape or form; from it's very first scene, in which Depp sits across from the legendarily weird Crispin Glover on a train, the movie seemed to confirm once and for all that the young actor was never going to let himself become your usual vanilla flavor-of-the-month.
Playing William Blake (the accountant, not the poet), Depp is extraordinary as a man with a one-way ticket to his own grave. He subtly navigates his anonymous middle-class hero through an accidental murder, a spirit quest, a bunch of not so accidental murders, and ultimately a communion with the fading soul of the country he never really knew. To this day, none of his other characters have ever experienced such a profound sense of change. For a character who's so often mistaken for someone else, William Blake feels as indivisibly real as anyone Depp has ever played.