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 & starts boiling away. At that temperature, the water will take any heat that your burner puts out and use it to turn into steam & 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.

At a high level, there's always a period of initial heating followed by a period of stable temperature. To be completely precise, the stable temperature is reached asymptotically, but in practice we think of it as preheating til close enough, then stable. When food is added, the temperature drops then recovers.

We tend to cook at a relatively stable temperature, especially for more sustained cooking, though there are certainly exceptions, where the goal is simply to reach a given temperature, and then you're done cooking. I'm not sure if these are what you're looking for, but:

• cooking meat - usually you want to reach a certain temperature for safety/doneness, and not overcook.

• candymaking - the goal here is to carefully heat sugar to a given temperature, and then stop as soon as you reach it.

• thickened sauces, puddings - starch and egg just need to get heated enough to set.

In your slow cooker example, yes, in equilibrium the temperature will be kept pretty much at 100C, since there should be at least a bit of liquid inside and it'll be at a very low boil. But it does take quite a long time to reach that equilibrium starting from room- or refrigerator-temperature food; sometimes people will even heat on the stove then transfer to the slow cooker.

It's pretty much the same on the stove. A given setting on the stove will translate into some amount of power. (You may not know what that translation is, but the stove is still producing some fixed power.) In combination with whatever's sitting on top of that burner, that'll result in some equilibrium temperature, where the power going into your pot or pan is equal to the power it's releasing into the air around it.

If you're cooking a pot of something with liquid, it'll take time to heat up, but once in equilibrium you'll presumably be simmering or boiling at a pretty constant temperature. If the liquid is thick or if things are stuck to the bottom, the temperature may not be quite constant through the whole pot, but it's certainly not just getting hotter and hotter over time.

If you're cooking without liquid, you generally find a setting that keeps things at roughly the temperature you want, and you generally preheat the pot or pan. You don't want it to take forever to cook, and maybe you want some browning but you don't want it to burn, so you get the heat in the right range. Again, as you add things the temperature will drop and recover, but you definitely don't want it getting arbitrarily hot and burning your food. So outside of preheating and recovery, the temperature will be roughly constant.

Even if you put an empty pan on the stove, and even if you crank the stove all the way up, it'll eventually reach some equilibrium temperature. It may have ruined your pan by that point, though.

As a dish cooks it will absorb less heat but lets assume that is constant

Let's say the serving temp is 160 F and burner is 200 F. If you left it on long enough the dish would get close to 200 F and over cooked. It will approach the (steady state) temperature in an asymptotic manner. (length should be temp on the graph)

In a crock pot it will come to cooking temp in an asymptotic manner and then hold that temperature.

Eventually, a temperature equilibrium will be reached, but how it will be reached is very dependent on what is on the stovetop.

1. A pot with water (or any other liquid that rather vaporizes than burns/pyrolizes) in it, and nothing stuck to the bottom, will stabilize at around 100°C, any further energy input will just end up creating steam. Same with a pressure cooker at 121°C (for 15 psi) if the mechanism that keeps the pressure works. Once all water is boiled off, see 2. If the heat input is too weak to support boiling, see 2a.

2. An empty pot, or one with solid contents or oil, will heat until either: a.) a temperature is reached where the temperature gradient causes as much energy to be lost to the environment as is put in, or b.) same as above, but before a stable point is reached, there will be a phase where contents pyrolize (consuming energy) or burn (adding energy) c.) the equipment fails from heat damage.

Solid matter/oil could vaporize too, but burning/pyrolysis tend to be more likely.