How long could a meat-based broth be held safely near 180F/82C?

Just to provide an official source, the USDA's National Advisory Committee on Microbiological Criteria for Foods (NACMCF) says this (unit conversion added):

Question 4: What minimum time/temperature parameters for hot holding would ensure food safety?

. . . For non-continuous temperature and time monitoring, a minimum hot holding temperature of 130°F / 54.4°C for a maximum time of 4 hours, based on information provided by FDA regarding the limitation of growth of Clostridium perfringens to no more than 1 log_10 in food, would be adequate to ensure food safety. In addition, the Committee concluded that a minimum temperature of 135°F / 57°C for a maximum of 8 hours, or a minimum temperature of 140°F / 60°C indefinitely also would be adequate to ensure food safety. . . .

You can read further details in the link and the detailed report which is linked on that page. But, basically as long as your food stays above 140°F / 60°C, according to this report, it should remain safe "indefinitely."

I would say that the main issue arises in the fact that you might add things to this endless soup/stock. It would only become potentially dangerous if the additions are below 140°F / 60°C when added (and thus cool parts of the food into the danger zone before they recover) or if the additions already contain toxins that might then be concentrated as the liquid evaporates off. Food above 140°F / 60°C should not be producing new toxins (from bacteria at least), but whatever is in there already may not necessarily be destroyed. Adding water though shouldn't pose a problem as long as it's heated before adding or is only added in a small quantity which wouldn't reduce the temperature below 140°F / 60°C.

My main other concern would be that any cooking vessel or utensils used be VERY non-reactive. Elevated temperatures will make it easier for metals or other substances to dissolve more quickly. A pot or pan or other vessel that might be fine for simmering for a few hours could leach more significant quantities of metals or coatings over days or weeks. If your cooking vessel AT ALL reacts with the liquid inside of it, this could pose a long-term issue.

(It's possible that some foods themselves might break down into less desirable components, but this will mostly be a quality issue than a safety issue at temperatures below boiling. At much higher temperatures -- which would be impossible in a liquid stock -- you could also potentially produce carcinogens over time as things break down, but this is probably unlikely for food kept in your proposed temperature range. If you were to let the stock reduce at some point to the point that it went past the demi-glace stage and became drier and temperature began to rise significantly, this could theoretically pose a problem. But in that case you'd probably get smoke and bad smells that would be a clue that your endless soup should be tossed because it had... well, met its end.)

I am trying to think of anything that would refute this notion, but I cannot -- there is no reason you should not be able to hold a liquid at those temperatures for an indefinite period of time. So long as you are reconstituting the mixture with water that is not contaminated, and doing so slowly enough to drop the temperature of the system below 130ºF / 54.4ºC, you should be able to maintain perpetually safe conditions.