What would cooking chicken at 140 degrees F for a prolonged period do?

Well, on some level, this is exactly what sous vide cooking aims to do--very gently bring the food up to the target temperature and no further. In theory you wind up with something that's more or less perfectly cooked all the way through with no part overcooked or undercooked. But of course this is typically done in vacuum bags in a water bath, which is very different from doing it in an oven, because the bag keeps all the natural juices in contact with the meat, preventing drying on the outside.

When doing this in the oven, it's unavoidable that some of the outside will get dry. Even at temperatures lower than boiling, some evaporation will occur and the longer you cook (a side effect of low temperatures) the more drying you can get. Beyond that, about the only downsides are that the fat renders very differently so the skin doesn't get crisped, and there's little carry-over cooking that goes on after the bird is removed from the oven.

I think it's definitely a matter for debate whether this kind of method results in better taste. By not getting things hot enough for the Maillard reaction (browning, basically) I think you're leaving a lot of flavor potential untapped. However, it's undeniable that, ignoring possible air-drying that happens, you'll have little to no risk of overcooking. That alone will increase your chances of a good result because overcooking has to be the most common mistake people make.

As to why you got your 140 degree reading when the meat near the bone was still pink, I think there are two things at work. One is that I understand that the meat near the bone is just naturally more pink and that it's fine to eat in that state if the temperature is right--there's some science behind it but I just can't find the article right now. The other thing is that there's always some variability to sticking a thermometer into a roast, and you might not have gotten your probe into the least-done part.

165F (73,9ºC) is the temperature set by the FDA so an idiot can cook chicken without getting sick. At this temperature, the chicken needs to remain at this temperature for less than 10 seconds so as to achieve an appropriate pathogen decrease.

140 (60ºC) is PERFECTLY ACCEPTABLE as long as the chicken remains at this temperature for at least 26 minutes.

The FDA and CFIA use the temperature of 165 because it doesn't require core temperature monitoring and timing. If you know what you're doing in the kitchen, like the OP does, 140 is perfectly acceptable... And in fact, produces the best chicken you've ever eaten in your life!

Source (provided by @roxr in a comment below):https://www.canr.msu.edu/smprv/uploads/files/RTE_Poultry_Tables1.pdf

Leaving food in the oven after it's finished cooking is generally referred to as "over-cooking" and for poultry in particular, is likely going to result in dried out and tough meat.

The chicken will not stop cooking just because you maintain the temperature. In particular, moisture will continue to be lost.

The "pink" you see near the bone (of, say, the leg), is a different type of pink from the raw meat. It may be due to the hemoglobin in the tissue which can form a heat-stable colour, and it is completely safe to eat.

(I am not a microbiologist but I've thought about this topic a lot -- do your own research if you really want to experiment and be careful)

I added a comment, but also, the FSIS (part of the USDA), had time-temperature charts: http://www.fsis.usda.gov/wps/wcm/connect/9ab2e062-7ac8-49b7-aea1-f070048a113a/RTE_Poultry_Tables.pdf?MOD=AJPERES.

They don't really want the average person using these, because you need accurate equipment. The 165F is the temperature that salmonella instantly dies (7 log10 die-off). Notice they never state that all salmonella is eliminated which is why you always need to handle left-overs properly. Note also that they deactivate at a little below 130F, so holding that temperature keeps them from multiplying.

TITLE: Modeling non-linear survival curves to calculate thermal inactivation of Salmonella in poultry of different fat levels AUTHORS: V. K. Juneja, B. S. Eblen, H. M. Marks JOURNAL: International Journal of Food Microbiology 70 (2001) 37-51.

These charts are for a certain humidity and fat content which is why the USDA says the 165F number.

For all I know, the meat could taste better the longer it is cooked. Does anyone know?

I have some theories and some ideas about this. I don't have references, but my understanding is that meat generally won't get tough if you keep the temperature low: sous vide style. The reason is that the proteins in meat denature as the temperature goes higher and when they denature, they tangle up and release water (very dumbed-down version). This is why they are tough and dry. The rate at which they denature is exponentially related to temperature (the same goes for the die-off of salmonella if you look at the time-temperature charts).

It's not as complicated as people are presuming. It works best with smaller pieces.

The 140 degree mark is the internal temp and not the cooking temp. Sous vide cooking has complicated this in some peoples minds as the cooking temperature itself is the same as the desired resulting temperature. This is not the same in an oven where you will need to cook hotter, and then reduce once desires temp has been reached, in this case 140f.

An oven at 250 will obtain this perfectly adequately. Just use a thermometer. Reduce oven to similar temps (not cooler) once the desired temp has been obtained and hold it there for at least 25 minutes.

The 140 must be maintained. it is the duration of time at 140 that kills bacteria, not the temperature of 165, at which point the meat is overcooked, and no potential pathogens remain.

With a meat probe that has low alarms as well as high, it is incredibly easy. your low will be 140. The high is only relevant to not ruining the meat at this stage.

The FDA Food Code recommends cooking chicken to 165°F (74°C). But the pasteurization of chicken is actually a function of both temperature and time. If you can hold your chicken at 145°F (63°C) for 8.5 minutes, you can achieve the same bacterial reduction as at 165°F (74°C). In his book, Kenji López-Alt uses this principle along with the principle of carryover cooking in large masses of meat to achieve succulent, perfectly safe results at a dramatically lower pull temp.

https://blog.thermoworks.com/chicken/chicken-internal-temps-everything-you-need-to-know/

Read the full article to understand the temp vs time. How much bacteria is killed at a specific temp and for what amount of time it would need to be held at that temp

It has been my experience that most smokers run at about 225 degrees F. At that temp it will most likely take a nice thick chicken breast at least 20 minutes to go from 140 to 145. That should be sufficient for health concerns and will prevent the meat from drying out unnecessarily. Also note that if you "glaze" chicken in a hot over after smoking it, you will ruin it.