The most efficient way to heat the water is with an immersion heater. (e.g. a Sunbeam electric kettle).
The outer surface of the heating element in an immersion heater doesn’t exceed 100[sup]0[/sup].
A well designed electric kettle has a plastic body and a small element, so as to minimise the thermal mass to be heated.
It also reduces the T[sup]4[/sup] radiation losses incurred from having the heating element at, say, 200[sup]0[/sup].
No question about it. If you already have one, you are better off using it. Also, for the purposes of cooking, I assume we want to bring the water to a boil but not really have it all boil away, so the question is really that of bringing the water to 100ºC.
Now, suppose the question is, is it worth for me to buy an electric kettle to save money? I tried to work out a realistic scenario:
Assume an electric kettle costs about the same as 250 KWh.
1 KWh = 860ºCL (H20)
Delta T (H20) = 80ºC
250 KWh = 250860/80 = 2687.5
With the cost of an electric kettle we can increase the temperature of 2867 liters of water by 80ºC
Let us assume we heat two liters of water every day for beverages. Depending on the % of losses on the stove, the breakeven period in years would be:
Loss
%
80 4.6
70 5.3
60 6.1
50 7.4
40 9.2
30 12.3
20 18.4
10 36.8
If you boil 4 liters then divide by two, etc. If the life of the kettle is under the break-even period, then you lose money by buying a kettle.
I just thought of a problem which is conceptually similar: When launching a rocket, is it more energy-efficient to go slow or to go fast?
This shares the same situation that you are injecting energy into a system whish loses energy over time. Just staying at a given point requires energy to replace what is being lost.
The answer in the case of a rocket is that the greater the acceleration, the more energy efficient. The best thing would be to fire the payload from a cannon at ground level. This is impossible for obvious reasons but rockets try to impart the highest acceptable acceleration because the shorter the time it takes to get out, the less energy you are wasting staying aloft.
An interesting difference though is that as time increases, while it takes more energy to keep the pot hot (because it is hotter), it takes less energy to keep the paylod aloft (because gravity is weaker).
The faster the better. That’s the correct answer.
So I post an answer which is totally different from what has been said until now and no one has anything to say? I guess everybody agrees with me. 
I just saw this, sorry to jump in late…but…Isn’t the most efficient way to boil water is with gas? Because to get the electrical coils and elements to heat the water, something has to create that electricity…Generally nuclear, coal or hydro energy used to turn those big turbines thus creating electricity. From start to finish it seems one would use a ton of energy through the inefficiencies of energy transfer (From the mine to the plant to the power station down the wires to your house into your heating element). Whereas gas is fairly efficient, little loss from start to finish. So, in those terms, wouldn’t gas be more efficient than that immersed coil?
-Tcat
Why not a solar concentrator?
Either that, or you’ve made everyone’s twit filter. But either way, Good Job! 
Actually, I don’t think the rocket is quite the same. Rocket’s have inertia, temperature doesn’t. Faster acceleration reduces your losses (due to Earth’s gravity being slightly reduced) while it increases the heat losses, due to larger temparature differentials.
OK, you mentioned that second difference. :rolleyes:
Nuclear would be more efficient than gas if you simply ran the water through the reactor.