While I'm blessed enough to have full functionality of both my arms, I have repeatedly run into situations where I am significantly more skilled than those I am playing with, and to keep things interesting, restrict myself to one hand when playing a number of games. While I am significantly better with both hands, it is not impossible to be somewhat competitive in many games with only one hand. Occasionally I've found myself (successfully) using these techniques in tournament matches when I feel a sufficient need to make a point. Moreover, I have in the past found myself with a pressing desire to play a new game, but absolutely no spare time, so I double-up eating with playing. For example, I beat (the gamecube version of ) The Legend of Zelda: Twilight Princess entirely with only my left hand, such that I could use my right hand to eat during my lunch breaks.
I figured I could give some advice as a result of my experiences.
First off, I should note that I primarily do this on a Nintendo Gamecube controller (as I am an avid Super Smash Bros Melee fan), but in my experiences they do translate well to the Xbox360 controller. I have not given serious consideration to a Playstation controller, and naturally this will not translate to keyboard/mouse.
Secondly, I should note that I've yet to find a good way (assuming an unmodified controller) to have immediate access to both sticks and all of the shoulder buttons simultaneously. Typically, you'll have to limit yourself to delayed access to something. This is a limited issue in some games (e.g.: fighting games), but can be very pressing in other games (e.g.: first person shooters).
You did not state in TFS which hand you have lost, and so I will cover both hands. There are two main grips I use, depending on the specific situation in the specific game.
The primarily left handed grip: Sitting down, place the controller on your left thigh or knee. Place your pinky on the left stick, thumb on the right stick, and pointer finger over the main part of the face buttons. To access shoulder buttons, have either your pinky or pointer finger reach over and around the controller to the appropriate button. This will most likely feel awkward at first, especially using the pinky on the left control stick, but I assure you with practice it is quite possible to become adept at it. The biggest limitation is the reach time for the shoulder buttons.
The second grip left-handed grip is a modification of the way the left hand typically holds the controller. I have often heard this referred to as "the left-handed claw". Instead of using the thumb on the left stick, slide it down to the directional pad, and use the pointer finger on the stick. If you try to also cover both left shoulder buttons you'll find you only have the pinky to provide support - rest the controller on your leg. The obvious limitation here is significant lack of access to the right side of the controller. I use this in SSBM for wavedashing when needed (jumping with up).
It is quite possible to switch between these two grips on-the-fly. While you'll have at best a delayed access to any given input device on the controller, you will have access to everything. With practice, I'm reasonably confident someone with one hand could progress through many 360/NGC games built with the intention
The right hand is, in my opinion, significantly harder to use, but not impossible:
The primarily right-hand grip: Rest the controller on your right thigh or knee. Place your pointer finger on the left stick, your ring finger on the right stick, and your pinky over the main face buttons. Like the left hand, reach over the controller with either your pointer finger or pinky when needing to reach the shoulder buttons.
I cannot think of any games that are completely playable with only the right-hand claw (see the left hand claw above for reference), so I won't really cover it. I should note that, when playing with both hands, I use the right-handed claw. However, most games that are okay with a delayed access to shoulder buttons could be playable with the primary right hand grip described above.
Whether the amount of practice needed to be proficient with the technique I've described above is worth it is up to you, but do believe me when I say that with practice it is quite possible for many if not most games.