Very many games use the nunchuk.
The Super Mario Galaxy games require the nunchuk; both are very popular.
Mario Kart Wii can be played with a whole host of control schemes, though I feel that nunchuk + Wiimote is one of the more accurate ways to play. One of the schemes is the standalone Wiimote (with optional steering wheel, but it's the same scheme).
New Super Mario Bros. does not require the nunchuk, and I believe there isn't even an option to use one.
Regarding your amended question, yes, I would strongly recommend picking up at least one nunchuk. Ideally you'll want as many nunchuks as Wiimotes, however one should be enough given the games they're interested in.
Super Mario Galaxy 1 & 2: 1 nunchuk max.
New Super Mario Bros.: 4 nunchuks max (thanks to Josh).
Mario Kart: 4 nunchuks max, 0 nunchuks min. Depends entirely on the way they like to play it, not worth getting more than one nunchuk for this straight off the bat. Not worth bothering with extra steering wheels either, in my opinion.