That can no longer happen (since January 1, 2014). Insurance policies now have out-of-pocket maximums thanks to Obamacare. The 2016 limits are: $6,850 for self-only coverage and $13,700 for a family plan.
So you would have stopped hemorrhaging money at $6,850, and then the insurance company would have been 100% responsible for the rest.
I realize that. I work in the health "insurance" industry (as a vendor).
Notwithstanding that, it was considered a voluntary procedure, though there was no way the stone would pass (11mm x 6mm) - average ureter size is 3-5mm and I had to sit in the billing office and work out a 'financial solution' with them prior to authorizing surgery (negotiating anything while in this much pain would probably be considered close to coercion).
As an aside, I really dislike "Obamacare", as a name. The Affordable Care Act. As an aside, a survey a few years ago asked self-identified Republicans this: "Would you prefer some form of affordable care act as an alternative to Obamacare?", to which many replied in the affirmative. There's too much potential for bias.
So you would have stopped hemorrhaging money at $6,850, and then the insurance company would have been 100% responsible for the rest.