Yes, but only with Gamecube controllers. Be sure to have an older Wii SKU, the newest versions removed the Gamecube sockets.

http://en-americas-support.nintendo.com/app/answers/detail/a_id/2790/~/how-to-use-nintendo-gamecube-controllers

I also think you can hack the virtual Wii in the WiiU and play Gamecube games with Gamepad/WiiU Pro controller/etc.

Yes it can play Gamecube games as long as you get a version that has Gamecube controller ports. Later revisions of the Wii removed these parts to cut costs to make a more budget friendly model (as if the Wii wasn't cheap enough at that point). Gamecube games operate in Gamecube mode, basically, and only recognize Gamecube controllers.

Yep. Under the flap is 4 sockets for GC contollers

Guess you'll need some GC memory too.

Yep, don't forget the memory card. Gamecube mode is isolated from Wii mode, so no saving onto anything but a Gamecube memory card. It was super convenient for people who already owned a Gamecube before, but at least you can get a brand new Gamecube controller at a reasonable price thanks to the Smash ones.

the older models can play Gamecube games. the newer models cant however.

Yeah, but you need a GC controller.

The first model can and it's even the best way to play gamecube games because of better video output quality, for PAL model anyways. Any gamecube controller will work, Wii controllers won't work.

They removed gamecube backward compatibility in the subsequent models of Wii unfortunately.

how can someone with 20,000 posts on a video game forum not be aware that the Wii plays Gamecube games? baffling

Yes but you need a memory card and a controller