Human beings are actually FASCINATING examples of PID controllers...especially as we learn.

I'm soon to become a dad for the first time and so my partner and I have become much more conscious and aware of babies around us and watching them learn things. It's really fun to apply control theory to that...

A friend has a young child who is learning to feed themselves. I was chatting with the friend while we drank beer and he helped the child eat dinner. using hands to pick up food (watermelon chunks) and bring it to mouth - very conservative and lossy PID-esque model of action, but you could watch small tuning improvements in real time. Then a spoon was introduced for sweet potatoes and the kid's PID controller clearly could not adapt to the change in mass in the system and we watched huge overshoots and control failures. Then it slowly improved...but isn't yet 'good'

then we tried to explain this to our partners (both reside in the 'S' in STEM)...

Can confirm. We use PIDs in espresso machines to control the temperature of the water used in brewing. Some machines have simple thermostats that over/under shoot ideal brewing temperature and a PID will help keep a consistent temperature or even let you override the temperature (if your machine runs too hot or too cold).

Do you have any insight on why PID controllers are used for this application? PID assumes a system with a continuous control input, but water boilers usually have an on/off switch. Is the input something like a PWM value?

My temperature controlled kettle does a way-more-than-optimal amount of switching when it's near the setpoint. That's consistent with using a PID controller into a PWM input.

Maybe there is a market opportunity for a water boiler that uses control techniques better suited to on/off inputs :)

A resistive electric heater can be PWM'ed to provide variable output. If you want to maintain a narrow temperature band, you can do that better by feeding the water temperature into your PID loop and have it control the PWM duty cycle, rather than to a simple on-off thermostat.

Yes! I use it to control both the amperage and temperature of a kiln.

3D printers use it to control bed and extruder temperatures.

Us Billies on the Hill use them in our stills.