There are one or two 'dones', depending on what you are cooking.
The first done has to do with food safety, i.e., the meat is cooked to a point where it is safe to eat. As Mark notes, with something like beef, say, a beef steak, it might be done at rare, medium-rare, etc., according to your taste. The outside of the steak (where pathogens are most likely to be) is at a much higher temp then the interior. The steak is safe to eat.
With burgers and other ground meats or meat products (like meat loaf), it takes higher internal temps to make the meat safe to consume - or somewhat higher internal temps that are held for a specific period of time (see
here).
For meats that are tough - like briskets, butts, ribs, and others - longer cooking is needed. Though these meats will become safe to consume relatively quickly - that is to say they will be pasteurized outside and inside, so, technically 'done' - they will still be too tough to make a pleasant eating experience. Done for these types of meats is when they have cooked for enough time to become tender.
For tender pork - like pork loin and chops, tenderloin, etc. - internal temps need only hit the high 130s-low 140s to be safe for consumption. The interior of the cut will likely be pink at those temps but the meat is safe to consume. HIgh finish temps for lean commercial pork cuts usually means dry, overcooked meat. Many people do have problems with pink pork, but this is usually a result of the erroneous assumption (that most of the country was under for decades) that all pork has to be cooked completely through to high internals to be safe. It doesn't.
Chicken presents a couple problems if whole. Chicken is safe as low as 155 (if that temp is maintained for a bit over a minute) or for practical purposes, 160. Though chicken breast should be tender and moist at this temp (and is where I consider breast to be done), it is not quite a high enough temp for most people for dark meat. Even though safety has been achieved (the juices will not run clear, btw, but the meat is safe) most people do not find the dark meat of chicken very palatable at these low temps, nor visually appealing (there will likely be pink - even red - near the bone, especially at the joints - and the meat might still have a rather pinkish cast). For dark meat most people prefer internal temps of at least 170 or higher. The trick with whole chicken is to cook the dark meat enough so that it is palatable while not overcooking the breast meat.
If the cooker temp is higher than the likely finish temp for whatever it is you're cooking, yes, the temps will continue to rise to a point. They can stall on their way up due to a number of factors, but they will rise, stopping only when they near the boiling point of water. Eventually as all the water evaporates and drips away they would start rising again, but at this point the meat would be well beyond the shoe leather stage.
