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.