A.ki.ra, as Tetsuo says it. My friend's brothers used to watch it when we were quite young. Needless to say, it was frightening and made quite the impression. That more or less set up my standards about what I want out of anime: visceral, tragic, superpowers, with a plangent note of hope.
That being said, it wasn't too much later I watched Sailor Moon (American version) and love it. So there's that!