What is the approximate size of the largest rocks that this tumbler can reasonably handle?

As long as you have the appropriate size and quantity of filler you could tumble a little smaller than fist size. It is important to emphasize that if you just add a bunch or one large rock(s) not much will happen. The rock(s) need to be tumbling over each other. This is where the filler comes into play. The filler could be Ceramic media or smaller rocks. We'd recommend you have a good understanding of tumbling before attempting to tumble larger rocks.