While it was puzzling that Nintendo elected not to provide backward compatibility for GameCube titles (although the system is certainly capable of it) talk is now suggesting that all may not be lost. In fact, Wii U owners may be able to play at least some GameCube titles on their systems.
Apparently, the plan is to make a number of titles available through WiiWare as downloads. Of course, you will have to purchase them, because GameCube discs are not going to be compatible with the unit. With the ability to support a number of USB storage options, downloading them to store will likely not be a problem.
Of course, Nintendo has not confirmed this to be the case; and in the past we have heard whispers of things like this only for Nintendo not to go this direction. While you would have to pay for the downloads, it does sound like a good idea to offer some of the best GameCube titles on WiiWare, and what company would not want to make money selling the same game a second time? We suggest that you file this in the plausible column for now, and wait to see how things sort themselves out.