  In this paper, we study the critical exponent of infinite words $\ubeta$
coding $\beta$-integers for $\beta$ being a~non-simple Parry number. In other
words, we investigate the maximal consecutive repetitions of factors that occur
in the infinite word in question. We calculate also the ultimate critical
exponent that expresses how long repetitions occur in the infinite word
$\ubeta$ when the factors of length growing ad infinitum are considered. The
basic ingredients of our method are the description of all bispecial factors of
$\ubeta$ and the notion of return words. This method can be applied to any
fixed point of any primitive substitution.
