This seems utterly terrifying but I'm curious if anyone has actually tried it and not died. If you tried it and did die, post that too.
I'll stick to sacrificing a nut and/or maillon if I end up in this situation.
From rockclimbing.com:
The Texas Rope Trick :
Please note that this can only be done if your height is 1/3 the length of your rope or less.
Directly anchor into the bolt with a sling so you can work safely, and clean your draw. Next, thread an open sling half-way through the hanger, so both ends are hanging even. Next, thread the tail of your rope through both ends of the open loop, stopping when it hits the ground. To avoid any confusion in the next step, you can rig your rappel device and just let it sit.
Now pull up the other end of the rope, and tie it onto the side of the sling with the bartacks. (This is just your pull strand once the rappel is done. This strand must not be threaded through your device.) You should now have three strands running to the ground. One tail is on the ground, it runs up and through both strands of the sling, down to the ground, then back up to a knot on the sling. Once you've double-checked that you're properly rigged, take up rope and weigh the device while you unclip from the bolt. Rappel down.
Now you want to pull the rope in such a way that you free the sling, rather than accidentally girth hitching it to the bolt. Gently pull the middle strand (not the tail end on the ground or the end knotted to the sling). The tail end on the ground will run up and clear the sling, which should now easily pull through the bolt. The faster you pull the rope, the more wear you're going to put on the sling. Due to this friction and the sharp radius of the hanger, the sling shouldn't be used for anything else. I'd recommend not trying this with one of the new ultra-thin slings on the market.
This is dangerous, and rigging it wrong could mess you up. If in doubt, leave the biner. You're trusting your life to a single bolt and the advice from someone from the Internet. Take it as you will, but I'm not responsible for your decisions.
Trad Texas Rope Trick:
Caution this can be very dangerous and should only be used by Diehard Dirtbags
Setup an anchor and attach to it.
Place one atomic bombproof cam independent of the anchor.
Girth hitch a trigger sling:
------------Single Stem; around the stem but above the trigger
------------Double Stem; around the trigger
Setup the "TRT" through the 'biner on the load bearing sling .[not the trigger sling!!!!]
Attach the pulling tail of the rope to the trigger sling.
Get on the "TTRT" and disassemble your anchor.
Pray
Rap down. [DO NOT RAP OFF THE TRIGGER]
Once down [one way or another] pull hard on the rope attached to the trigger sling.
The whole kit will come down, there is no need to pull rap side rope at all.
Use at own risk. May cause severe injury or death.