Is there a wi fi booster that works best with Airport Extreme?

If the connection between base stations will be wireless, then only another AirPort Extreme or Time Capsule would work. However, if the walls will introduce too great of a level of Wi-Fi interference, this option will not work very well either.

However, you are still left with two choices:

1. Use an Ethernet connection between base stations. If that isn't practical or even possible, then
2. Consider using PowerLine adapters to create a pseudo Ethernet connection using your home's electrical circuit.

We have thick plaster and wire mesh walls

This is unfortunately a worst case scenario in most homes for wireless connections, as you already know.

need to boost the signal.

If you plan to connect another AirPort Extreme using a wireless connection back to the "main" AirPort Extreme, the AirPort will not really "boost" the signal at all. It will "repeat" the signal, meaning that it can only repeat the quality and speed of signal that it receives.

But chances are, the WiFi signal will have already slowed down and weakened significantly by the time that it reaches the location of the repeater. So not much will be gained by adding the repeater.

For that reason, it is easy to see why you would want to avoid trying to use wireless to connect the repeater if at all possible.

If you can use an Ethernet cable to connect the remote AirPort back to the main AirPort, the remote AirPort will receive a full speed signal to broadcast, resulting in a much stronger and faster WiFi signal for that area of the house.

Any universal repeaters/extenders will work. It should say universal.

For example the Netgear EX series.

https://www.netgear.com.au/home/products/networking/wifi-range-extenders/default.aspx

There are a number of consequences of using these types of products.

Most serious is low speed due to double hop. And from my experience unreliable operation long term.

They also do have network issues caused by the way they operate.. ie as a client to main wireless router and then effectively working as completely different router. They hide all clients connected to the extender which cannot then be controlled from the main router.

If the only option you have is wireless I would recommend replacement of the airports with mesh products. Apple never got to the point of developing mesh but it is very popular now in large homes or homes with high loss.. ie concrete, double brick and your sort of hard plaster walls.

Using ethernet or in some cases powerline is preferred.

The cheap solution is usually the worst here.. so if you own the place spend a bit of money and do it properly with ethernet.