  This paper provides a systematic study of fundamental combinatorial
properties of one-dimensional, two-sided infinite simple Toeplitz subshifts.
Explicit formulas for the complexity function, the palindrome complexity
function and the repetitivity function are proven. Moreover, a complete
description of the de Bruijn graphs of the subshifts is given. Finally, the
Boshernitzan condition is characterised in terms of combinatorial quantities,
based on a recent result of Liu and Qu. Particular simple characterisations are
provided for simple Toeplitz subshifts that correspond to the orbital Schreier
graphs of the family of Grigorchuk's groups, a class of subshifts that serves
as main example throughout the paper.
