Along with birth and death injury is unavoidable for characters in most RPGs. Unless you play Paranoia in which case you play, die and bring out another clone. Characters are going to get hurt. For the purposes of this article we're dealing with bodily hurt. Though the affects of social and intellectual wounding could make for an interesting future post.
Classic Traveller had a simple take on wounding. Minor injuries could be healed with three days of rest and a first aid kit. Serious injuries took a hospital or sickbay to fix, otherwise you were basically crippled pending proper care. Beyond serious wounding you were dead.
Now Risus lets you specify what happens to a defeated foe. Maybe you shoot the gun out of their hand, maybe you shoot their hand, maybe you plug them between the eyes. We dealt with being dead already. Now let's deal with healing.
Many heroes have heightened healing, whether it be the conventions of the genre or some mutant/genetically engineered/nanotech gimmick. They don't necessarily have to go to a hospital but consider, regeneration won't help you with everything that can go wrong. For starters getting your head knocked off will probably have you reaching for a new character sheet. Likewise grow your own bandages won't help you with drowning unless you get resuscitated. Getting a limb dislocated is also not going to benefit from regeneration. Also foreign objects in a body may be problematic after the skin closes. You didn't remove the bullet before the hole healed? How do we remove it? Surgery? Better be quick.
What about more conventional healing processes? Back alley clinics (cyber or any other flavor) are a staple of some SF settings. Sure you can buy a cyber arm there cheap but what was it used for before you got it? Maybe the hidden pistol in it was used in a crime and the cops have some bullets from it in forensics. Maybe the chips you buy are special prototypes that were not recognized but the mega corp that designed them had bounty hunters on their trail. Maybe the chips are wanted for all manner of sensitive information they contain. If you buy a chip that lets you compute orbital trajectories and get a chip with all manner of secret FTL routes then odds are someone is after you (especially if FTL is unknown.)
In a horrific vein we have organ transplants. What happens in your world when a kind man gets the heart of a killer? The hands of a strangler? The eyes of a god? The opportunities are endless.
In the fantasy genre we have clerics handling the healing and being regarded as hit points in a can. What if your deity doesn't want you healing non-believers? What if he wants favors in exchange? You don't want to know what that demon wants in exchange for patching you up. John Constantine had a whole story arc on this in Hellblazer. What if accepting healing from a deity slowly brought you around to their way of thinking?
A final thought tying some of this together an experimental treatment facility is a great group origin for characters. It doesn't have to be SF or even modern. Mages and clerics do have to come up with new spells somehow and practice casting them. They might not like word of their mistakes getting out. Their mistakes might be your new adventure group.