Somewhere in the mid-1990s my sister took me to see ‘Natural Born Killers’. They were letting people older than 18 into the movie. She was 17 and I was 14, but they let us in anyway. We felt like we had hacked the system. 
It was certainly very counter-cultural for me at the time, plus a great soundtrack (NIN!). Maybe also Pulp Fiction at the time with its non-linear plot.
A few years later Pi and Eraserhead were so counter-cultural for me. The first time I saw both was at a joint screening in a small cinema in the attic of some community centre, so the setting already gave the impression of something small, unpopular and counter-cultural. Plus Pi has an amazing soundtrack, through which I got to know Ae and Afx.