Taipei is game-crazy. From pinball machines to toy dispensers, the gray city blocks are sprinkled with the neon glow of mini arcades, with enticing names such as “Happy Moments” and “Sweet Box” (?).

Whether to kill time, add some spark to a date or cap off a boozy night out with friends, claw machines have become a mainstream pastime in Korea in recent years. For many brave players who step up to ...