Does the temperature increase the longer the food is on the stovetop?
In your oven, or any cooking method which has a thermostat marked with temperatures, the heat source is designed to turn on and off as needed to keep your food [somewhat] constant at close to that selected temperature.

On the stovetop, with say the burner on medium heat, the heat source just keeps putting heat into your food at a steady rate until you turn off, or adjust, the burner. However the temperature of your food doesn't just keep increasing forever, because when you cook on top of the stove in a pot, you usually have some water in there with your food.  When that water gets up to its boiling temperature, it can't get any hotter. It starts sucking up heat to become steam &amp; starts boiling away. At that temperature, the water will take any heat that your burner puts out and use it to turn into steam &amp; leave the food. So your pot stays at that temp, about 100 degrees C. until all the water is finished converting to steam, and the contents of your pot are dry, and free to begin increasing in temperature beyond that boiling point. Pretty soon after that the food starts to scorch and burn.

So to answer your question, yes the temperature levels off at the boiling point of water, but not because your stove guarantees it, it's because the water surrounding your food guarantees it. After your water boils away, the temperature of your pot is free to increase until it matches the temperature of your burner, at least on the bottom.