When braising, how deep should I fill a single pot?
If you are simmering, boiling or poaching something, the depth does not matter. As long as your food is completely immersed in liquid, all the liquid has a sufficiently regular temperature due to convection, and it cooks well. So the stews you mention are no problem, you can fill the pot up to the brim. 

Braising is a different beast entirely. In braising, you want the liquid to reach to the half of the food or a bit lower. Obviously, you can't do that if you stack your food. When you braise, the heat penetrates into the meat from all sides, but much quicker from below than from the sides (which are only steamed). The upper half still gets done early enough because of heat conducted from the hot bottom half. If you put a piece of meat on top of the first layer, 1) a much larger height (100% of it) is getting heated by steam only, not liquid, 2) the boundary between the two pieces of meat slows down conduction considerably, and 3) the heat which gets transferred bottom-up is much lower, as it is in contact with a piece of meat and not the pan bottom. So, a piece of meat on top will get much later to the desired temperature. 

You can get away with it, if you are not after perfection and if you are cooking for long enough for both layers to cook through, but the degree of tenderness will still vary within the stew. So, if you are braising large chunks of meat or vegetables, do a single layer only. 

If you are cooking a rice dish, or something with finely diced ingredients, it is again OK to make it deep, even if you are not covering them in liquid. In this case, you'll want to stir them a few times during the cooking. Don't stand there stirring all the time, it's counterproductive. Do it 2-3 times during the whole cook, but do it thoroughly, exchanging the "layers" as good as you can.