Are there any Wii games that require me to use a Classic Controller and won't work with a Gamecube controller or Wiimote+Nunchuck? Are there any controllers that have both a Wiimote and a Gamecube plug so I don't have to buy two similar controllers?
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.
I don't think so either. But I'm pissy about MegaMan 9 and Samurai Shodown - they support classic but not GC controller. It was exactly these types of games I bought the GC controller for. On the other hand, Mario Kart and Smash Brothers and virtual console games all support GC controller, which I like.