How many times can you reuse bones to make broth?

Beef bones can be used multiple times, but less flavor and gelatin will be extracted from each additional use. Harold McGee's "On Food and Cooking" describes this.

Because a standard kitchen extraction of eight hours releases only about 20% of the gelatin in beef bones, the bones may be extracted for a second time, for a total of up to 24 hours. The resulting liquid can then be used to start the next fresh extraction of meat and bones.

I would imagine chicken bones would be similar, although, since they are smaller, the first batch may be more effective than 20%.

You can use chicken bones to make broth only once, all the goodness gets cooked out of them the first use. You could re-cook them for hours and get nothing from them.

There's a specific term for reusing ingredients for stock twice: remouillage (which literally means a "rewetting"). Usually this "second stock" is not used directly for broth, as it has significantly less flavor than the primary stock. That said, depending on the type of bones, the amount of meat used in making the stock, etc., it may still have a very pleasant (if lighter) flavor.

In traditional French cuisine it tends to be used as a cooking liquid to make a new stock with (that is, you might cook chicken #1 twice, and use the second stock from chicken #1 to make a richer stock using the bones from chicken #2). It can also be reduced for a glace, in which case the significant concentration will make it taste a bit more flavorful.

I personally tend to do this frequently when I'm making stock and save the "second stock" for miscellaneous uses, like a cooking liquid for rice or vegetables, or as the basis for a future broth.

Generally, doing a third (or more) use of the bones for broth will extract very little flavor, mostly only giving you a bit of the remaining gelatin. Any flavor that does still exist will also become increasingly unbalanced.

I just pressure cooked a turkey, with roasting in the boiler just prior to the initial pressure cooking session, then afterwards, pressure cooking the carcass (bones and remaining skin and unused meat) two more times.

After the second pressure cooking session of the turkey carcass and remains, the turkey bones were easily broken apart, either broken in the middle or the ends of the larger bones could be somewhat broken open. I pressure cooked one more time, for a third time. I cannot recall the condition of the bones afterwards, but everything fit nicely within a sieve afterwards.

I'm starting to look at this question from a different angle, if you were held within a prison camp, how many times are you going to cook the carcass for nutrients? I think we can cook the carcass as many times as we want, the bones will likely just liquefy at some extent, but is this material (calcium) desirable or pleasant on the pallet? Likely not. I think performing enough cycles for the bones to start showing weakening, and upon weakening, break them open and process them one more time might be best?

The flavor of a good broth is gained by the liquid extracting the flavors and substances from the bones over a period of time. If your stock/broth has decent flavor, then, naturally, the bones should have little to none left to give any subsequent attempts at extraction.

Ideally, the answer should be "none" if you've made a great broth or stock on the first try.