The Combination:
Up - Up - Up - Down - Down - Down - Left - Left - Right - Left - Left - Up - Up - Down

There are two ways to put this code in, conveniently there are two control schemes so here's what you need to know:
- Keyboard:
Use the arrow keys, your character will do a little dance as they move but when you're done, their head will swell up.
- Controller:
Use the D-Pad, your character won't move, you'll just be cycling gadgets, when finished, your head will swell up 3 sizes that day.

Notes:
Any time you have to continue from being taken out by a bad guy, everyone's heads will go back to normal. This is normal as far as I can tell, and only makes you have to put the code in again.

The title says that it only works in New Game + or after the main story is completed, if anyone else gets the code working before finishing the game, let me know and I'll change that, I may or may not have been clumsy when I tried entering the code before beating the game.

If you decide you've had enough of the ridiculous proportions of the Arkham Characters, simply put the code in again and everyone will be as though they put some ice on that crazy headache they collectively had and the swelling will go back down.

DOES NOT WORK ON EASY. ONLY NORMAL OR HIGHER DIFFICULTY.

Here's a video that someone else made demonstrating it:

I'm not trying to take credit for the video, only raise awareness of the presence of this hilarious way to beat up villains. Re-watching the video there's a pretty big issue in the video itself, the guy has a different combination shown in the video, this is incorrect. The combination shown in this guide is from the description of the video and works (tested/verified)