>>... if their liver became damaged beyond usefulness,
Actually, the liver *can* regenerate already - the only one of our visceral organs that can do this. It only can't once it's shot through with scar tissue (sclerosis) which blocks regeneration. Scarring (as opposed to regeneration) seems to be a mamalian evolutionary 'choice'. We don't grow limbs back, just the liver (again, so long as it isn't sclerotic).
That mentioned, assuming we're limiting ourselves to eucaryotes, planaria (flatworms) seem to be effectively immortal - they 'reproduce' by grabbing something and tearing themselves in half. Each half then regenerates the 'missing' other half. Michael Levin's team (Levin Lab at Tufts) has done some fascinating work with them - and they're not the only ones.