My current CJ has been my daily driver for 6 years or so. It has never left me completely stranded, or needed a tow. But I always keep a basic set of tools and common parts under the rear seat.
But some common breakdowns I had:
Lots of carb/fuel issues, like a stuck choke or clogged jet. Nothing as fun as rebuilding a carb, or cleaning plugs on the side of the road. It was years before I worked all those bugs out. Had a fuel pump fail, so I ran my jerry can on the hood, gravity fed. Ran out of fuel more times that I can count.
Lots of ignition issues. I had a Prestolite that the PO swapped in. I fought with that system for years chasing gremlins, randomly killing the engine. Eventually gave up and did the HEI route. I still have a "backup" ignition button that bypassed the prestolite harness, and added extra grounds everywhere.
A few drivetrain issues. I've blow a few ujoints, which is pretty scary. It makes a hell of a noise, but generally no big deal. I carry spares, and driven with just the front driveshaft before. Blown a rear wheel bearing, but drove home (5 miles) making a skreech that probobly woke the dead. Had a tranny fail, it was stuck in 3rd gear! Drove 15 miles home... oddly, not as hard as you might think.
I've had 3 different clutch failures. (throwout, master cylinder, and hose). Every jeep owner should be able to drive indefinetly without a clutch though i think its a lost skill with the younger generation. Likewise, I've had a few total brake failures. Fortunetly, I've always kept my e-brake in working order, so I was able to get home... though I stuck to backroads.
Lots of coolant issues: Lost of the fan belt a couple times. I'd drive a mile, pull over to cool down, drive a mile, etc until I could get to a parts store. I carry a spare now. One winter, I had too much water in the radiator and it would not circulate right, causing violent overheats. Its crazy to have STEAM flood the cab in sub-zero weather. I've also overheated from an intake leak, and a head gasket fail.
Oh, and lots of crashes. I drive it year-round, and despite the "4wd" sticker, its not awesome to drive in a heavy snow or icestorm. 3 different times I've skidded into guard rails, or trees, or rocks. In all the cases, the battery violently busted its bracket, and smashed the heater coolant hoses. Fortunetly, those are easy to bypass with minimal tools. I've lucky the battery never bashed the distro, or alternator. it has a beefier bracket now though the frame itself is pretty warped from the crashes.
But never been stranded at least.