Here is a kit that lets you build your own tiny game console. The TinyScreen Video Game Kit lets you play games such as Flappy Bird or Space Invaders with the Joystick TinyShield. It comes with a TinyDuino processor, TinyShield USB board, TinyScreen OLED display, joystick, and 140 mAh battery. You also get basic instructions to get started fast.
The video game kit is available or under $75 at this time.