I failed to figure out that you can stomp things with pressing down while jumping, so I beat around half a game without killing anything except bosses(with throwing rocks and eyeballs attack). It's pretty hard to avoid damage and preserve health, but it's actually a fun way to play this game. It's a shame that apparently it's impossible to beat it like this - I completely stuck before finding the frog friend and had to watch some random longplay and slap myself in the face.