Fortunately, this is a company-owned quad, so I don't have to trouble shoot it, but thought some might find this either familiar or of interest...
the machine is a 1995 trx300 4X4, which was purchased used from an area dealer. It has started and run reliably for over a year. It doesn't get used much, but I make sure it gets started and driven around the back lot where it is stored every 2 weeks to keep the battery charged and fluids circulated. So, last week, I started it and it fired right up, ran normally for maybe 20 or 30 seconds, and then the starter re-engaged, and could not be shut off. Neither turning off the key, nor sliding the kill switch worked...the plugs stopped firing, but the starter just kept cranking the engine until I disconnected the cable at the starter. Anybody else ever see this happen?

 

I haven't, but it sounds like the starter solonoid is getting a false signal to operate, or the solonoid itself is bad.

 

Most likely the solenoid like IdahoVinnie said. Its the only thing that could get power to the starter w/ the key off. If you push the start button w/ the key off, nothing happens. Take a screwdriver and jump the solenoid post w/ the key off, starter will spin. Only other possibility is a short somewhere that is bypassing the ignition and starter switches and taking battery power to the solenoid to activate it. Easiest way to determine if that is happening is just unplug the solenoid connector from the harness and if the starter still spins, its the solenoid itself.

 

Makes sense that it's on the battery side...although, having messed with Fords from 2N tractors to EB's (Early Broncos for the uninitiated-the FUN ones), I've never seen a solenoid go bad in that manner before, so it kinda blew me away.