It boils down to the question how dense you can encode, retrieve and write information- and this can be very dense.

Conways compression comes to mind, where a simple timing description and some start states can be used to describe a very complex outcome.

The problem is, that for every function computing a irrational number, you got to have the output as input again.

Imagine if it where otherwise, a completely lightweight PI-function, traveling for eternity, for which nothing in the universe could determinate if its going in a loop or going towards a circle constant.