yetanotheruser vs its_prolly_fine, 26 July at 02:49 UTC: 255 consecutive draws.
Two hundred and fifty six rounds. Two players throwing the same hand, over and over, neither willing to be the one who changed. It ended when scissors finally cut paper.
WAIT! The counter broke after 255. I think we got to at least 300 (before its_prolly_fine decided to end it).

