You will most probably overload a circuit breaker before overloading the socket. If so it should blow.
I believe a wall socket (single or double since it uses the same single connection at the back) is expected to pass about 15A but I am prepared for the electrical experts to correct me on that.
At 230v this would be about 3.45 kW. You may add up the wattages given on each of your son's appliances to see if they add up to more than that.
That said it isn't good practice to put break-out boxes in series, but four loads from a double socket shouldn't be a problem.
Recall that the gas boiler will take very little as it is just using electricity for running its electronic control circuitry, not heating the house. The freezer may spike due to induction, but is not permanently taking power: it goes on for a short while then off again. The tumble drier also is only taking power when on. Lighting is minimal.