Another solution might be counting something (slots in a fan-like disc on a shaft, a paint spot, teeth on a gear or sprocket, etc) using a PIC, and reporting it via a serial A-to-D.
Ten turn pots are the neat solution, if you can get enough resolution. By the way, is there a restriction on how small your gears have to be ? Driving a pot probably doesn't require steel gears 1/2" thick. You probably have servo motor gears you could use.
If your are not to worried about precsion (only need a few bits), pots are made that have no stop. The resistance is not complete in the small angle between the ends of the resistance element. You'd have to read the resistance, and also count the number of times it went through 255.
A caution: Three turn pots are available: they have a 1/4" control shaft that becomes thin, and the thin part drives 3 balls around in a race formed in the cover (a rough and ready planetary system), on the side away from the mounting bushing. They are used for calibration adjustments (eg, in pH meters). I suspect that they might not maintain a linear relation between angle an resistance (they might slip
This doesn't matter in a pH meter where the operator just keeps turning until the effect is right.