The funeral happens once. The panikhida is what the Orthodox tradition does instead of leaving the matter there. It is a short sung service, a compressed version of the burial rite, with the same litanies for the departed, the same prayer for a place of light and refreshment, and the same closing "Memory eternal". Because it is short it can be asked for at any time, and because it is asked for by name it becomes the ordinary way a family goes on doing something for its dead.
The days are fixed by custom rather than by doctrine. The third day after death, the ninth and the fortieth are the standard sequence, then the anniversary, and after that whenever the family wishes. The forty days carry a widespread popular understanding of the soul's passage after death which the church has never defined dogmatically; the services are kept whether or not any particular account of what happens in that period is held.
Alongside the services for individuals, the calendar sets aside several Saturdays of Souls, when the commemoration is general. On those days the priest reads out the names submitted by the whole parish, sometimes hundreds of them, and the dead of that community are prayed for together rather than one family at a time. That is the balance the practice keeps: a rite personal enough to be requested for one named person, and a calendar that makes sure the ones nobody requests are not left out.