Why isn't the beef fat in my chili melting? I'm making some Texas-style chili ("beans optional" :) for the second time, and the fat on my brisket just does not want to melt. As the recipe instructs, the chili sits on the stove for 4+ hours at a simmer.

My somewhat-trusty instant-read thermometer says that the temperature is around 205 °F after about 3 hours of simmering. As far as I can tell, very little fat from the brisket has actually melted/rendered by now. I noticed a similar issue last time I made this chili as well: the final outcome had some rather large pieces of fat which were not a particularly nice texture. Here's what it looks like so far:



The brisket is already very tender, and there's very little fat floating on the surface, but there's significant layer of fat still on the beef chunks.

What's going on? I'd really think that multiple hours at 200+ °F would melt most or all of the beef fat in the pot. Is it too hot? Not hot enough? Do I have any hope of getting the fat to melt so I can either skim it off, or mix it in?
Not all the fat in beef (and generally, in any kind of meat) is the same. 

Any piece of meat will have a certain amount of marbling fat, which is intramuscular and is hard to remove, intermuscular fat and subcutaneous fat, which can be trimmed off. 

From your picture, it seems to me that the fat covering the outer layer of the beef chunks is gristle, which you can easily remove after having cooked your Chili. It is however hard to remove by increasing the temperature, as fat molecules are generally quite big and they are not soluble in aqueous solutions, which makes them kind of float on your sauce in an emulsion. 

From my experience as a cook, the best tip I can give you when it comes to Chili or stews is to choose a lean piece of beef, with as little as possible of visible marbling fat and before cooking, to remove as much of the visible fat as you can possibly do. Cooking it gently for four hours will make sure that the meat remains as tender as you can wish. :)