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A period of a single cell would take 35328^2 periods and (2048*2048)^2 cells of the most underlying game of life simulation.

A Life acceleration algorithm would help with that, but each cell would still require 4Tb of storage (though that storage could be shared by all cells, II think).

https://en.wikipedia.org/wiki/Hashlife

And there is not real purpose since the rules are the same for every level. Making them different and more complex would be interesting.

Once you know how to simulate GOL within GOL, you can go as deep as you want.

It's turtles all the way down: https://www.youtube.com/watch?v=D6aP9S9rEQk

According to the description, this was done with video editing, rather than an actual running nested simulation.

It is a simulation of Game of Life in Game of Life. They did loop the video though. Obviously it's not possible to fully simulate such a huge machine. You could do it with sufficiently clever optimization, but the result would be equivalent to video editing.