Assuming there are at least 3 alleles of the gene $G$ in total - $G_R$, $G_S$ and $G_P$ - is there any gene for which the following is true?
$G_R$ is more dominant than $G_S$.
$G_S$ is more dominant than $G_P$.
$G_P$ is more dominant than $G_R$.
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A quick search gives this same question in this Reddit post.
Apparently, there is not yet an existing example of such dominance of three alleles on one another.
That said, if you're interested in rock-paper-scissor patterns in nature, then you will be interested in the side-blotched lizard. It has three genetically encoded male "sexes", that also determines their behaviour. At a population level, the three sexes follow a rock-paper-scissor pattern of successful sexual competition, in that orange-throated individuals can outcompete yellow-throated ones, which can outcompete blue-throated ones, which can outcome orange-throated ones. A simulation has been carried out to model the underlying genetics, but that model doesn't involve a rock-paper-scissors pattern of genetic dominance.