Reading glasses just because the more places I have reading glasses, the more I can lose them or forget to bring them along and still read. Sometimes, I need to -- beer menus, things like that.
Picks, because I sometimes lose or forget them, too, but also because sometimes someone else needs one. Nice to have spares.
Ibuprofen because I play hard. My doctor tells me two or three, taken an hour before a gig, will keep the swelling down. Also, sometimes I get headaches and I don't want one to cloud a gig or a jam for me. Ditto for my band mates. "Got any brown vitamins?" is a common enough question.
A pen and little notebook. You never know when someone might ask for contact info.
Strings because sometimes, they break. Fiddlers moving to mandolin forget this, because fiddle strings don't. Bowing a string just doesn't produce the wear-and-tear on a string that plucking it does. When I do gigs with Ron Sommers, he'll bring a second guitar. He breaks strings once or twice a night; rather than change strings, mid-set, he just picks up the other guitar and replaces the broken string while they're teaching the next dance.
I own a string winder, but it looks like I've either loaned mine out or left it somewhere. I don't change strings often enough to absolutely have to have one, but Ron makes excellent use of them.
Mine, wherever it may be, also has a wire cutter on the far end of the handle, which lets me trim the string end once it's on. Other people quickly run the leftover end between their thumb and a hard edge, like the back of a pocket knife, which makes the end coil up, getting it out of the way until they have a chance to trim it.
A shaker egg. Vergil Weatherford convinced me they're great instruments. Besides that, if I'm at a party and some non-musician's feeling left out, I can lend it to him. Mine's shaped like a little Seckel pear so I own two of them, of course, but I only carry one.
A tuning fork. Everyone else carries around electronic tuners, but I still carry an A fork and tune by ear. Most of the time I use "festival tuning": I just tune to the folks around me.
I hear musicians complain, surprisingly often, that using an electronic tuner has decreased their ability to tune by ear. I have no idea whether that's actually true, and I'd be interested to know whether anyone's actually studied it. It may simply be that they can no longer ignore small differences in pitch. Musicians who try practicing with a metronome marvel that the metronome isn't keeping steady time.
If you are tuning by ear, by choice or necessity, tune facing a wall. It lets you hear yourself well in noisy environments. If you've never used this trick, try it at the next music party you go to. I'll do it leaning the top of my head up against the wall, so it catches all the sound. It's a great trick, but doesn't work well with instruments that are either hard to move around, like pianos, or un-tuneable, like hammered dulcimers.
Q: "How long does it take to tune a hammered dulcimer?"
A: "Nobody knows."
Q: "What's the difference between a Harley-Davidson and a hammered dulcimer?"
A: "You can tune a Harley."