I think it's worth noting that the system cannot work with an even number, so you're stuck with 3 unless you want to go up to 5 or higher, or you're willing to allow ties.

How about... animal, mineral, vegetable? Animal eats vegetable, vegetable... grows on mineral(?), and mineral bludgeons animal. It's (perhaps) a recognisable combination from games like "Twenty Questions" as a quick way to narrow down what someone is thinking of, but I think this would be a unique application of it.

Have you seen these bigger versions of Rock Paper Scissors? Perhaps you can pick out some logical, interesting, and unique combinations of three out of one of the bigger charts.

PS: Some might find it interesting that I originally conceived of and prototyped KarBOOM as a physical Rock Paper Scissors game, hence the "Sapoi" mutator ("sapoi" is what we called Rock Paper Scissors in school in Fiji).