Banish Misfortune (Varnish Me Foreskin) - I've known this tune for a long time, and I never seem to get tired of it.
Morrison's Jig and The Butterfly get played a lot, but I am starting to get tired of them.
Only three - sheesh! We do loads of others. It's rare anyone can get folks to learn something new, but the most recent hit is Showman's Fancy. (I think it's the same as Bill's Showman, with a whole bunch of triplets in the second part.)