When making a thick simple syrup (1 part water, 1 part white sugar), how can I know when I've cooked the sugar for long enough? Is it possible to overcook this?
You aren't actually trying to cook anything. When you heat a solution, it makes dissolving a solid in to that solution much easier. So you can dissolve more sugar in to hot water than cold water. With a 1:1 ratio, you wouldn't be able to get all the sugar in to solution with cold water. So, you heat the water to allow more sugar to become part of the solution. So the answer is, as soon as there is no visible sugar in the water, you are done.
If you were to cook it for longer, you would reduce the syrup, increasing the concentration of sugar in the solution. You'd have to reduce it an awful lot before you burnt this. However, if you reduce it too much, then the liquid will become solid when it is cooled. Both of those would probably qualify as "over cooked".
Here's a nice science description of sugar solubility. Interestingly, they say that sugar has such high solubility in water that you can get 1800g in to 1L. Using this volume to weight conversion site, that's approximately 7.5 cups of sugar in 1L of water. 1L = 4.2 cups, so you can get ~1.8 Cups of sugar in to 1 cup of water (under ideal circumstances). So, if you are actually interested in cooking your water off to get a more sugary syrup, you can't reduce the water to less than 1 / 1.8 = or 55% of the original volume. However, if you do want a ratio higher that 1:1, then I would suggest introducing the appropriate amount of sugar in the first place and stopping when the sugar is dissolved rather than using less sugar and cooking the liquid down, as the first approach is much more precise.
(for purposes of this answer, I have ignored the possibility of super saturating the syrup, as that would be essentially useless for cooking....although it would be funny)