Do some Wii games require a Classic Controller?

Solution 1:

No, there are no games that require a classic controller but do not work with either a gamecube controller or the wiimote & nunchuck combination.

There are games however that only support two out of the three combinations, most notably, Monster Hunter Tri, which only supports the classic controller and the wiimote+nunchuck combination, but do not support the Gamecube controller

Do note however that some games are not optimized for the gamecube controller and play a lot easier on the classic controller (SNES games notably) and the gamecube controller is the only controller that allows you to play gamecube games.

Solution 2:

No. To the best of my knowledge, a Gamecube controller will work for all of the games that state that they require the classic controller; if you already own a Wavebird or Wired gamecube controller, you do not need to purchase the classic controller.