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Question #1: Can we get 3 kilowatts of power out of some lithium ion batteries?

There are high-current versions of lithium batteries. Conveniently, they're widely available because they're used for vaping.

I found a battery that looks reasonable (https://www.18650batterystore.com/products/molicel-p42a). Its stats: 4200 mAh capacity, 3.6V nominal, 45A continuous discharge, and retail cost $4.99.

At 3.6V and 45A, each battery should output 162 watts. Rounding to 150 watts, we'd need 20 of them to make 3 kilowatts. So $100 worth of batteries.

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Question #2: How long will they last? Long enough to boil water?

At the 45-amp discharge rate and with 4.2 amp-hour capacity, it should take 4.2/45 hours = 6 or 7 minutes to discharge them.

By my math, it takes 335 kilojoules to heat a liter of water from 20°C to 100°C. A 3 kilowatt kettle should be able to do it in 335/3 = 112 seconds.

So the batteries should be able to boil water around 3 times before discharged.

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Those calculations are for running on battery alone. Since you can get 1500W out of an American 120V outlet, you could make a kettle that draws 1500W from the wall and boosts it with 1500W of battery power. (I'd use two heating elements.) Then you only need $50 of batteries.

The kettle is going to be a bit heavy, though. The batteries are 70g each, so 20 of them is 1.4 kg. Also, I don't know much batteries heat up when cranking out 45 amps, but I bet the answer is a lot, and you may need active cooling and/or thermal shutoff.