You know, I jump in this thread all excited, ready to make a Huniepop joke...

... and ya'll have beat me to it ;)

When you said games that you play with one hand I thought you meant something else

GOOOOOOOOOOTHIC!!!!!!

He can also play games at the Nutaku website with one hand. Although it might feel pointless if the other one is broken.

Only if you're extremely masochistic.

Well, there is the Xbox Adaptive Controller, which could be used to make some games accessible to more players.

Also, hardware remapping might be useful in cases where software remapping is not an option.

One example of remapping: Many games only need one joystick and the face buttons. In such cases, if the controls are remapped so that the right joystick controls movement (instead of the left), the game might become one-hand accessible.

Control redundancy can also work. Dragon Warrior 7 (PlayStation) is an example here; the left shoulder buttons can be used instead of the right face buttons (plus the game is (mostly) turn based to begin with). VVVVVV (available here on GOG, and which I already mentioned) is another good example.