On my accordion, I can play Fairest Lord Jesus, Be Still My Soul (aka "Finlandia" by Jean Sibelius), Come Thou Fount of Every Blessing, This Land is Your Land, and the most hummable parts of Beethoven's Jesu, Joy of Man's Desiring. Both melody and chords.
The key of C major is working best for me right now, partly because of some unfortunate sticky black notes in the upper register. What can I say, it's a $50 accordion. Also, it is a Major Diatonic instrument, which means that the left hand chord keys are basically arranged in a handy I-IV-V configuration, with the keys of G, C, and F represented. That means it lends itself most happily to music in one of those three major keys with simple harmony. Minor tonalities and jazzier progressions are possible, though, but not yet. I've got minor keys figured out strategically, but they are not tactically practical at the moment (which means I can't play them yet).
Now one might think that being limited thusly would be a drag, but it's really not. Just because you've got the 16 Crayola box and not the 64 Crayola box does not mean you can't draw some great pictures and do some serious coloring.
On the list of things that are more difficult than they sound I now include the song Greensleeves, because of the minor key, and that boom-chick polka rhythm that everybody and their uncle Gunther from the old country does...can't figure that out yet, either. I think there's a trick (uhm, I mean technique) to it. Also, long, slow lyrical phrases don't come easy, but they don't really come easy on any instrument.
At the moment, I don't own any zydeco music, nor do I have any recordings of Astor Piazzolla, the great Argentine tango composer. I shall have to remedy that soon. By the way, I just learned that Astor Piazzolla's middle name was Pantoleon. Translated, that means "lion pants", I'm pretty sure. How cool is that for a name? Anyway, he's supposed to be the Chuck Norris of accordion music, so I need to get up to speed.
And check this out:
Accordion Hero II is just being released, too, I understand. It includes M.C. Hammer's Can't Touch This and also Ride The Lightning by Metallica. Does that rawk or what?!?!