A random thought: can you shake an egg vigorously enough to beat it inside its own shell? Aside from this patent I can find nothing about this. Any thoughts, recipes?
Someone made an instructable on how to make what they call a Japanese Golden Egg (although a search for that just gave me lots of hits on Angry Birds); they spin the egg quickly, breaking up the internal structure, and then boil it to get a scrambled-egg-colored boiled egg.
There are a few devices out there. Most involve poking a tiny hole in the shell to insert some sort of needle on the device which breaks the yolk. That seems to be the main challenge, rather than creating the speed necessary to mix the white and yolk components.
The infamous Ronco offers a $25 item that doesn't seem to have a needle, but its reviews are rather poor.
You might try the paint-shaker machine at a home improvement store; it'd be a fun experiment if you didn't mind cleaning up a few messes. :^D
It is unlikely that simply shaking by hand will beat an egg. Eggs have inner anchors called chalazae that affix the yolk to the membrane. It is also cushioned by the membranes and an air pocket.
Anatomy of an egg: http://www.exploratorium.edu/cooking/eggs/eggcomposition.html
Yes you can! This video shows how to securely attach a raw egg to a power drill, then use the drill to scramble the egg inside the shell.
The egg is secured (with the long axis of the egg perpendicular to the drill) with tissue paper and wire inside a cut-off soda bottle, and a bolt is put through the bottle cap to be held by the drill chuck. The drill is run alternately in both directions.
I just vigorously shook an egg (by hand) and when I cracked it, it came out like nothing had even happened.