What is the purpose of this spiral wire on top of the transformer body?
It's probably a hand-wound inrush/current limiting resistor.

Basically, it's a big, low-resistance power resistor used to limit the inrush current when the device is plugged in. 

Since this is part of a battery charger, and not a device with significant internal capacitors, it's probably more to protect the rectifier in the case of an output short then limit inrush current.