Successfully complete World 1 to unlock an option for the Staff Credits mini-game at the Options menu.
World 9
Successfully complete all eight default worlds without using any continues to earn a crown in each of them to unlock the ninth world, Sinking Swamp.
World 10
Successfully complete World 9 without using any continues to unlock World 10.
Super characters
Successfully complete World 1 through World 9 with only one character (either AiAi, Baby, Doctor, GonGon, MiMi, or YanYan) to unlock the "Super" version of that character. The "Super" version of that monkey can accelerate slightly faster.