Is there an upper time limit when simmering stock?
You could implement a perpetual stock, where you keep it simmering indefinitely and renew it by adding fresh ingredients over time. The tradition of keeping a perpetual stew boiling for weeks or even months dates back centuries, and was a way to keep the ingredients from going bad in days before refrigeration. 

https://alehorn.com/2016/01/07/is-medieval-perpetual-stew-for-you/

If you just want to simmer a single pot of stock indefinitely, you can do that safely by adding water on a regular basis in small quantities (small enough that the stock temperature does not drop below 140 degrees) but if you're not using some and replacing the ingredients, over time (several days) the stock will eventually acquire a sort of acrid taste (speaking from personal experience.)

I think the issue is that there's a limited amount of the aromatic compounds which produce the desired flavor, and over time, these tend to evaporate. Replenishing water doesn't replenish the flavor. The smell of stock simmering means those aromatic compounds are now in the air and no longer in the pot. Eventually, the amount of nice flavoring compounds leaving the broth to the air exceeds what you can extract from the remaining bones etc. Oxidation of the remnants leads to a less pleasant flavor. There is also the question of the pot itself adding metallic flavors if the stock is at all acidic.