In addition to Xbox 360 and Xbox One controllers, it was discovered that Steam Link also supports PS3 and PS4 controllers. I decided to give my Wii U Pro controller a shot. :
Amazon.com: Steam Link : Video Games
How to Use Xbox 360/One Controller on the Steam Link - VGU
Steam LinkとXbox 360 Wireless Controllerの通販 by やぎお's shop|ラクマ