Atlantis No Nazo (Mystery of Atlantis)(NES, 1986)

Partially non-linear (multiple level exits, some doors warp you several levels ahead - SMB, can't tell which door leads where)
One hit deaths and death pits (some pits warp you to hidden levels - finding them is a matter of trial & error)
Some permanent abilities can be gained however they are optional (shoes which let you stand on clouds, bulb - see in the dark briefly after using a bomb, S/Super Bomb - smart bomb, mic - lets you speak into the 2nd famicom controller's mic to freeze enemies, star=permanent invincibility (!) besides from your own bombs and pits)
​No continues or save feature (100 levels in total though it's unlikely you'll visit all of them)
No map feature
​Can open doors and make hidden doors appear with bombs but not break walls or other obstacles
Your own bombs can kill you
Time limit in each level (clock power up - slows down timer)

Video Review

Influenced by: 

Influenced: