Why is this obvious? And why do you want to leave it on permanently? These are hardy devices, but trying to keep it on indefinitely will eventually cause some harm, as with any device. but with the arduboy, the harm is particularly to the screen, each of those pixels is an LED, and if you leave it on as long as you possibly can, they will eventually burn out and/or become dim and dulled, greyish instead of white. This isn’t something that’s easy to do, however, and can only really happen if you actively try to leave your arduboy on for as long as possible. The battery should give you a decent few hours gameplay at the very least, but if you truly intend to leave your screen displaying something, I would try to minimize the amount of pixels being lit-up on screen. Still don’t recommend it.
TL;DR: Don’t leave it on permanently. It will mess up your screen.
Whenever you select a game from the FX menu, what’s happening is that it’s taking the .hex file for that game which is stored on the FX mod chip (the smaller black IC right between the DOWN button and the A button), and flashing it to the bigger chip in the corner, the one with a circle of smaller pcb components around it. The bigger chip is the ATMEGA32u4, the brains of the system. When you upload a sketch from your PC, it uploads directly to the ATMEGA32u4 and it will stay there until another game is uploaded, either through FX or PC. To access whatever game is on the ATMEGA32u4, simply turn the system on and you will be greeted with the “Arduboy FX Loader” screen, where you can mve left or right to see categories. Don’t move left or right. instead stay on the Loader screen and press either A or B. Whatever program was last uploaded to the ATEGA32u4 (whether it was from FX or PC doesn’t matter) will be started.
If you want to permanently add a game to one of the categories, so you don’t need a PC to play that game next time, follow this guide. It is rather technical, but should still be easy enough if you try your hand at it.