Best answer: "I don't know. Probably very bad for you. And they'd be very cross with you."
If the fear of disgruntled CERN researchers (what's the Greek "-phobia" word for that, I wonder) isn't enough to make you keep your hands to yourself, rest assured, nobody thinks such an adventure would work out for you, health-wise.
One of the scientists interviewed here mentions that particles traveling close to the speed of light start emitting synchrotron radiation, something he calls "very nasty". (Note the fun look of terror in his eyes that goes along with this pronouncement.)
And with good reason. Consider Anatoli Petrovich Bugorski, a Russian researcher who accidentally stuck his head into the proton bean of a synchrotron in 1978. The good news: He survived. The bad news: The left side of his face swelled up and peeled off. In the long term, Bugorski has had to deal with hearing loss, fatigue, facial paralysis and seizures.
Conclusion (at least, until somebody tries it on Mythbusters): Sticking any part of your body into the LHC is probably a bad idea.
Thanks to EricHarley for Submitterating!
P.S.: Only the first five minutes or so of this video are dedicated to the LHC question. After that, scientists answer questions about the multi-verse and the threat posed to us by nearby stars going supernova. Enjoy!
