Speed Reducers as "Speed Increasers"

Yes, but it gets harder and harder to turn the input as your ratio increases. You'll basically get the same torque ratings as when it is used as a reducer (reversed, of course), but think about what that means: high input torque, low output torque.
To get the high output speed requires a large torque, but a slow rpm at the input. Your input would have to be something that can handle that. You couldn't just hook up an ordinary AC motor rated at the gearbox horsepower. You'd have to include some type of high-torque soft-start circuit or use a DC motor that can handle the high starting current. This is easy enough to do in an industrial setting, but if you're just fiddling around it could get expensive.
Don Kansas City

I have been working on a similar problem. I'd just like to see how fast I can get a set of gears going. The moment of inertia makes it hard to begin turning the input shaft. It seems that you need some sort of clutch or transmission to start a subset of 3 or 4 gears going, then increase the speed until you can add another speed-increasing gear to the front end. I haven't found much in the way of automatic transmissions that aren't installed on cars.
I am also having trouble setting up a breadboard for these experiments. Have you found a suitable breadboard (if you are doing this on your own) or setup to align gearheads, gears, motors, etc. to try out your ideas? I am having trouble setting one up that is fairly stable and precise, yet adjustable to many arrangements.