Mighty No. 9

This is fairly straightforward. You have to complete a level without touching any items that restore health (including the ones that Patch delivers to you) or using the "back" button to refill your health. This means that you cannot do it on the City level, as there are unavoidable healing items right near the end, just before the boss.

I did it on Power Plant, while trying for "Unbreakable". I can confirm that dying does not affect this achievement because I messed up big-time fighting Dyna and died on my first attempt, then unlocked it on my second try, so go ahead and set your extra lives to 9 and just let yourself die if you need to practice. Obviously it helps if you can make it to the boss untouched (or if you are better than me and can easily complete "Unbreakable", then you are golden. Just don't accidentally touch a health-up during the level).