Every gig I do these days is improvised, so we never know what we're going to play. It's all off the tops of our heads, so Beatles tunes naturally get thrown into the mix.
Crowd-pleasers are Twist and Shout and I Saw Her Standing There, but they're boring to me.
Personal favorites are the ones that are fun to harmonize on: I Should Have Known Better, Chains, Nowhere Man, That Boy. We usually perform as a two-man band, but I use a harmonizer box onstage so I can sound like two to five voices. Two to five voices that, even if they don't get the lyrics right, are still in agreement with one another. (The five of us kill on Eagles tunes, e.g. "Take it To the Limit" and "Take it Easy".)
I used to do "Let it Be" a lot when I still had my singing voice, but nowadays I lack the control and range to do it well. "Oh! Darling" I still do, but only at the end of the night when I won't need my voice again for a few days.