Forget anything with batteries - we have a set up that came from Amazon where the bell units plug into a wall power socket.
We've just moved - at the previous house we had 3 bell units (one on each floor) and 1 button, here we have 2 bells and 1 button - you can get various combinations with multiple buttons and bells. Just need a battery in the button push unit(s)
Our previous experience with battery powered bells was poor . . . . and one of the neighboiurs had a car key fob that chimed our doorbell from time to time - presumably as it scrolled the frequencies one matched our door bell!
Rob
I would also avoid battery powered radio bells. The battery in the push button bit is fine, as it is only connected when the button is pushed. The radio receiver has to stay on all the time.
There is no such thing as a battery powered receiver that has to stay on a long time on a small battery, while having a fast response. It simply takes too much power.
Mobile phones don't last that long when turned on, and they do clever things to only turn on every couple of seconds for a really short time. As long as the cell tower knows where they are, it can accept a call on the phone's behalf and wait the few seconds until the phone turns on and responds. Cordless are similar.
Car locks use the 15 kg car battery and then only last a month.
Home heating systems with motorised valves only turn on the receiver every 30s or so, which is fine for the radiators but would be a bit tedious for a doorbell.
TVs etc have two power supplies. One is phone-charger sized, and runs the receiver for the remote, and that turns on the serious power supply when needed.