You don't really need to use a whole chicken to make broth, that's kind of a waste imo. Instead, if you de-bone and remove the skin from the chicken parts when you get home from the store with them, and gradually save those up over time, those are all you need to make the broth. In fact, you can even skip the carrots and celery and expensive fresh herbs such as thyme, and make a really simple broth with just onions and garlic, salt and pepper, and perhaps an acid of some sort, i like to use a tomato but a little wine works too, which helps draw more flavor out of the bones. For the vegetables, I just save up scraps in a bag in the freezer, and use those - such as the base and leaves of celery, the tops of tomatoes, the bottoms of carrots, etc...
It does take a long time to make a pot of broth over the stove, but at the same time, you aren't really doing much with it throughout the day except occasionally stirring it. Otherwise you can make it in a slow cooker if you don't like leaving your burner on, or a way faster alternative is to make it in a pressure cooker - which only takes about an hour and a half and extracts a ton of flavor.
The website Steamy Kitchen recently published an article where she makes a bone broth in her slow cooker for several *days* - in fact she recommends cooking it for at least an entire day before eating some of it, then she adds a little water each day after eating some of it each day, to keep it going.
