I mention Impulse Labs and their battery-assisted 120V high power induction range in the comments elsewhere. Seems like a similar concept could be used to make an incredibly powerful kettle; throw in a battery that charges from the mains, and when you ask to boil, send in 20kW and boil the 250ml in 4 seconds.

    4.18 J/g/C * 250g * (1/ 20,000 kJ/s) * 75C = 3.918s

For that order of magnitude to work, in practice, the most challenging aspect will be getting enough surface area between the water and heater.

Otherwise, you will very quickly vaporize the water near the heater and the resulting lack of contact will inhibit heating the rest of the water volume.

Yeah that's a great way to start a fire with a lot of steam first :)

Microwave radiation could work to transfer heat in even as boiling initiates in spots. All the best kettles are BIS dual use items.

If you can transmit that amount of heat that quickly, I think it'd be much more convenient and feasible to have it in the form factor of an instant-boiling-water spout next to the sink, rather than a kettle. Then, rather than having to fill the kettle first, you directly get the amount of hot water you need into the vessel you want it in, and you can put a precise amount of heat into a precise amount of water flowing through a pipe to emit it at the right temperature.

By the way, you can already have a boiling water tap today, you just buy a device that uses hot water tank to store the energy you rather than the battery. Insinkerator sells these. It might not be as energy efficient as the hypothetical tankless water boiler as described by you, because you have some losses from the heat slowly leaking away from the tank, but given the battery costs, I suspect that over the lifetime of the device, these losses add up to less than what battery costs.

Yes — in fact this is a problem with the high-power setting on our induction hob (which I think is something like 5kW). Liquids bubble vigorously before they're even warm.