Not all of these are "great" as far as technical analysis is concerned, but these are some films that have had an impact on me and which I think approach greatness in their ways (rather than just being personal favorites). In no particular oder:
Arsenic and Old Lace -- I saw this for the first time as a kid, and it was the first time I grasped the concept of dark comedy. Happily it stands both the tests of maturity and time.
Tampopo -- Probably the "greatest" movie on this list. Also, marry me, Ken Watanabe?
Paper Moon -- Two words: Daddy Issues.
Smoke Signals -- The first time I'd seen a non-stereotyped, positive, and accurate portrayal of Indians in the media.
Hedwig and the Angry Inch -- How a musical about an East German transexual punk rock singer so perfectly captures exactly what it is like to be an 18 year old American girl, I'll never understand.
The Apartment -- One of the few romantic comedies (dramadies?) that gives me hope.