These two have been mentioned, but i just wanted to give my "endorsement" of them since both are must-haves for any Kermit collection in terms of how they represent Kermit's character. In fact i'd go so far as to say that these two deserve to be assoicated with Kermit just as much as Bein' Green and Rainbow Connection...
This Frog from Sesame Street - to me, this just always seemed like the ultimate Kermit Song
The Christmas Wish from John Denver/Muppets - not only is this SO Kermit, but may indeed be one of the very best holiday songs ever written - it's a song of good will regardless of the faith or belief system of the listener - totally universal in its worldview, and i'm proud that Kermit is the singer.