Here's what you can do with two AR pattern rifles, some hydraulic brake line, several beers, and time on your hands. Think of it as the AR 2-step.
Here's what you can do with your 10-22 if you fancy the roaring 20's look or the WW2 GI fashion.
Which makes me wonder if one couldn't mount multiple 10-22's in a row, with the bolt of the first connected to the trigger of the next down the line. Each pull of the first trigger would fire the succession of rifles one time each, and depending on the delay from one gun to the next, a modest repetition rate on the first gun would be indistinguishable from a really high-rate machine gun, although no gun in the assembly fires more than once from a single pull of the trigger of the first gun.
Any semi-auto rifle with an exposed bolt would work for this, and they don't even have to be the same make or model, although some would empty their magazines sooner than others.
Somebody build one of these! Or if you have 3 or more vaguely similar .22's, send them along to me, and I'll put it together in my copious spare time.