The last case like that I remember was a girl who had a catastrophic accident during surgery and ended up braindead. Her parents sued to keep her on life support though there was no chance of her waking up.
Yep that's Jahi McMath. She received a tonsillectomy and extensive mouth surgery to treat her severe sleep apnea, and then had a major bleeding event which rendered her brain-dead.
That's actually a very messy case, which I've read heavily about. For one, the family decided to bring Jahi a cheeseburger after her procedure, even though solid foods were obviously prohibited. But the hospital also didn't properly respond to the fact that Jahi was in an extreme amount of pain afterwards, and they didn't even notice she was bleeding out until way after she went into cardiac arrest.
It's further complicated by the fact that a law firm is currently milking the family dry by filming videos of Jahi's random muscle spasms and bolstering their belief that their daughter, who has extensive and irreversible brain-damage confirmed by dozens of experts, actually has a chance of recovery.