So today, I went and got new ties on the ole xj rig. The tire shop couldn't get my lug nuts off, so when I got home, I hammered a smaller socket onto the stripped lugnut, with some super glue in it... using an extension, I took my trusty extendable ratchet, and tried to turn the socket. To my surprise, the driver on the extension broke off of the ratchet, and is now stuck in the socket, which is stuck to the lugnut, which is stuck to the car. Where do I go from here?

 

What kind of a "tire shop" was this?

 

 

Front or back? Steel wheel or aluminum?

 

I had this problem on my TJ - two different tire shops couldn't get the nut off and the last took a torch to it and damaged the rim. I ended up just drilling out the wheel stud. It took some time but the tire came off.
Since then I make a point to tell any shop that is removing the wheel to be sure to only torque the wheel nuts to spec and not a hair more so I can get the tires off if I need to.

 

What do you call a mechanic that finished last in class at tech school? A mechanic.
Many many very good techs out there, but there are also some duds.
You may have to try drilling out the centre as suggested.
If your nut and socket are exposed and not deep in a pocket of your rim then:
Alternate, get a good sized nut (3/4"?, get a handful) hold it in place at exposed end of your socket, mig weld in the centre to fasted the nut to the socket. Wrench on the welded nut, best if still hot. It may break off. Weld on another and wrench on. Heat from the weld will penetrate and perhaps all will come off together.

 

Like peep said, weld something to the extension that is stuck in the socket. Or... maybe a pipe wrench or a chain wrench? With any of these a piece of steel tubing 4-6 feet long can be used to slide over whatever tool you use as a cheater bar for leverage. The stud may break but you will have to change it anyway so no big deal.

 

As was suggested to me on another thread.
One should deflate the tire for max safety.