At least I seem to remember the character having its leg cut off and then grabbing it (not that it would be of much help continuing the game with a leg less, though).

Well, they are doing amazing things with medicine now a days. :D

I guess

I know a game where you can wear your own severed head as a hat.

Neverdead, maybe?

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There was a DOS game that came out in the early 90s, with very crude VGA graphics, called Executioners. It was a beat-em-up that mimicked Final Fight and Streets of Rage. Anyway, the boss at the end of the first stage was a zombie whose attacks consisted of grabbing hold of you while puking on you, throwing his own head at you, and ripping off his own arm to beat you with it (and then resocketing it).

I just checked. There doesn't seem to be any rules against it in cricket.

Ditto for basketball.

Dwarf Fortress, if you don't pass out from pain and/or bleed to death.

Dwarf Fortress? I've never heard of it specifically but limbs can be lost and treated as items.
*EDIT* Gah! Pipped at the post! Good job Darvond, and I see you know Avernum too.

Man I'm stumped... Perhaps I dreamt it? None of these seem to fit :\ I'm not surprised at all that in Dwarf Fortress you can do that, but I've never played the adventure mode more than a minute or so. Could be one of the other ascii games I used to play though, like Nethack / Cataclysm : DDA.
I think I'll wait for 2 or 3 more guesses and then simply give the solution to the game that seems closest, which at this stage is Die by the Sword, though I could swear it was a more recent game :P

First one that came to my mind. I even beat the final boss with it. Turned out to be pretty hard since it has relatively low stats, despite me being way overleveled by that point.

hmmm, yeah, it's like...on the tip of my brain

some FPS...

damn...

Not your own limb, but zombies in Terraria can sometimes drop zombie arms, which are really good early-game swords.

Through three states?