Is there any danger to letting food cook in a slow cooker for a very long time?

It's safe. All that matters for safety is that the food stays out of the danger zone (above 140F).

But it sounds like a pretty reliable way to overcook things. Perhaps that's why it sounds absurd to you? Slow cookers tend to be somewhere between simmer and light boil (probably at least 180F), and there's very little that won't be fully cooked after half a day at those temperatures. If you cook for days, you'll start turning beans and vegetables to mush, and may manage to make meat tough from overcooking. So at best it's pointless, and at worst it's going to mess up your food.

I often cook pork shoulder steaks at 60°C (140°F) for 48 hours sous-vide and as your shoulder is an intact piece of meat you really only need to be worried about any bacteria on the surface of the meat; presuming the shoulder is submerged in liquid, a cooking time of 24 hours at 60°C is long enough to pasteurise the surface and the interior. There's a good thread on eGullet that, while about sous-vide, is applicable here.

Also, in the massively unlikely scenario that the pork is infected with Trichinosis the USDA guides state that holding the meat at 140°F for 1 minute will kill the parasite

The main purpose though of cooking your shoulder I presume is to tenderise it and at temperatures of 140°F you won't get the characteristic falling apart texture associated with braised meat.

What you're suggesting was effectively a form of food preservation in medieval times -- just keep the food warm at all times.

This works best if you keep adding something to it (not just liquid), so there's something that hasn't completely turned to mush ... and you might want to hit it with a shot of vinegar or citrus to perk it back up when serving. You can also add acid while cooking to slow down the breakdown of potatoes and onions (and maybe other vegetables), but in the long-cooking you'll lose some of the brighter notes.

If you try it with stew, you're going to end up with something closer to ragù by the time you're done. Personally, I like that in a pot roast, but I know some people aren't fans.

You could also add some fresh, mostly uncooked items when serving, that just need to be warmed through. (eg, fresh or frozen peas (not canned), been sprouts, or some diced bell pepper or onion, as appropriate for the dish).

Perpetual soup, which is exactly what you're talking about here, was a staple in many old world diets. It's still done in many places around the world today. Poland, Alaska, Russia, and many of the colder environments where it's harder to get food in the winter times. There is a risk to all things cooking no matter the preparation you take.

Perpetual soup is actually less meat and vegetables and more the scrap of what you've already cooked. The bones and perhaps edible organs. If doing this in a crock pot it would be best to simply let it cook, adding water as needed.

I do this myself, my crock has been going for almost a month now. Mind you I empty it into a roasting pan clean it and then add my mix back once per week. All things considered, I've never had any problems with flavor and the broth is amazingly nutritious.

The only real issue I've seen, from my own mistakes, when I tried this a couple years ago was adding directly on top of what is already in the pot. Anything you add needs to be under what you've already cooked and only when you're done pulling from it for at least four hours. It takes time but eventually you can get down a schedule for adding to it that works for you.

The technical answer is, as long as the food stays above 140 F / 60 C, pathogens are not going to grow. This is the hot equivalent to refrigeration. Note that there are probably no studies of holding foods at these temperatures for truly extended periods, but you pays your money and you takes your chances.

On the other hand, even with the lid on, over time, water is going to escape, and the food will begin to dry out. Even slow cooking meats like pork shoulder or ox tails can and will become overcooked and tough over time, even if there were no moisture loss.

With the fringe exception of a stock pot that is continually replenished, I cannot think of a single good reason to do this.

The USDA recently lowered the recommended safe cooking temperature of pork to 145°F (60°C), from the long-time standard of 160°F (70°C). However, this really applies more to lean cuts like loin - for a bigger cut like your shoulder, 165-180°F (75-80°C) is probably a safer bet.

Cooking at that temperature for 24 hours should give you no problems; do test it with a probe thermometer to make sure it is the right temperature all the way through.

190°F (90°C) would have been slight overkill even before the change, and 275°F (135°C) is just crazy.

As for the original discussion, I wanted to add something: If you're going to do this, first test your crock pot's settings. Fill it with lukewarm water, then put it on warm, for instance, and check every hour for the temp. If its lowest setting gets higher than 140F (60C) in less than two hours, it's completely safe. On warm, mine is about 180F (about 80C), so it's fine. Low and high bring it up over 200F (about 95C), the only difference in the settings is how quickly this happens. Last but not least, for the most safety, I would briefly microwave/ bake/ saute any new solids going into the pot over time, and bring any new liquids to a simmer (microwave's fine), so that they don't drop the temperature of the crock pot. If it's a big piece of meat, just get it to room temperature (thaw in the fridge first if frozen, then half an hour on the counter), and sear the outside well before adding it. I actually have an "endless soup" going now, because of some issues with my fridge. I don't like and can't afford waste, but also can't safely store leftovers right now, so I decided to keep things hot in my crock instead. I made an initial pot of soup, and then we ate most of the solid bits with a little of the broth. I added some more water and let the little bit of leftovers sit hot overnight. The next day, I put in new meat and veggies at the time I would have put it in, if I were making a fresh crock pot meal. I also add more/ new herbs, because the scent of the herbs breaks down after being held hot for a long time. The older items do turn into a sort of mush, but they make a very rich base for your new items, which won't be overdone if you do it this way. I actually like it so much that I may keep doing this, even after the fridge is fixed! lol That said, I personally will change my "endless" pot after three meals. After all, it could end up tasting quite same-y after a while. When you work in food service, the rule is generally not to keep prepared items for longer than three days under refrigeration. Obviously, we keep things for longer in our refrigerators at home, but it's a good general rule of thumb if you're concerned about safety. And, discarding a little leftovers every third day is better than discarding leftovers every day, or leaving them to while away in your fridge, be forgotten, etc.

To the guy with the crappy crock pot: New crock pots are not the issue. You got a cheap crock that's probably made in China, or some other place where they don't care. Get something better. I have a newer crock pot from around 2014, and it has never given me any off flavors. This is not a new/ old issue. This is a good ceramic vs bad ceramic issue. My crock pot was about $40. You don't have to break the bank, just don't get a skeezy off-brand cheap-as-dirt model. Get a crock from a reliable brand name. Mine is a Hamilton Beach. It's not the best pot ever, the rubber seal actually takes some heat damage if it's run on the high setting for too long, but said damage is superficial, not functional. Rarely sticks, can be cleaned with an SOS pad without taking any damage at all to the finish, and not a single off flavor. I actually bought this new pot because an OLD crock pot was given to me, and I was tired of that weird taste you're describing. It was very slight, but it bothered me.

Pork needs to be cooked throughout. It needs to be above 72°c for most bacteria to die. As long as you have increased the heat and let it alow cook above that head for a bit you will be fine. Remember that these heats are just guide lines but if you are serving it to other people then it is probably best to adhere to them.

With a new Crockpot, I did this, and the food has taken on a chemical flavor from the pot, which is nasty. Sort of like overheated heat in plastic when it contained BPA's. Even water cooked for a day or so, takes on the taste.

I think there's something unstable in the newer crockpot finishes for multi-day cooking. Those who don't easily smell things won't notice it. Those with a good nose, won't be able to eat the food.

And I am not sure it should be eaten with that awful taste in it. Be careful. Older crockpots didn't do this.