Perfect Blue (2001)
Paranoia and schizophrenia visually personified.
Presentation:
For an anime film to make it to this list, it would have to be truly spectacular. I chose this over Paprika because it may be slightly easier to understand, though more difficult to digest. The cinematography is one of the most extreme and dysfunctionally confusing experience by design. The way to story is presented truly gives you a glimpse into a psychological nightmare. I wouldn't be surprised if this film alone has influenced the Safdie brothers.
Story:
The story is utterly insane, open to interpretation and you really don't know what is real. There are so many wtf moments and the fact that you can't discern reality and what is fake just makes everything more terrifying. It may change how you see human perspective, trapping you in this 2 hour solipsism nightmare.
Conclusion:
This is probably one of the most psychological experiences I can think of that could change how you view cinema. No other film can make you feel this paranoid. It's very rare for such powerful films like this to exist, so take care to maximize your first viewing. This could be the textbook definition or reference if you wanted to experience insanity. A chilling and potentially life altering experience.
Recommendations
If a cinephile could have a Princess Bride.