Seeing as Hodor's current chirper activity is spouting movie quotes for the memes, I'd say the type of game show he'd be invited to would have something to do with popular movie quotes. Perhaps you have to complete a quote like with hangman and incorrect guesses means you have to run laps through an ice cold slime machine or something. That seems on par with Japanese game shows. I don't think Hodor would do well with that kind of game, he's not super well versed in popular movies or television shows. And even if he was, it's not like the contestants or judges will understand what he's trying to say. The best he can hope for is Hodor-specific quotes from Game of Thrones, then he'd be the best man for the job. Hodor doesn't try and hide who he is, so if people in Japan love Hodor's chirps they'll love him on a game show.