  Rich words are characterized by containing the maximum possible number of
distinct palindromes. Several characteristic properties of rich words have been
studied; yet the analysis of repetitions in rich words still involves some
interesting open problems. We address lower bounds on the repetition threshold
of infinite rich words over 2 and 3-letter alphabets, and construct a candidate
infinite rich word over the alphabet $\Sigma_2=\{0,1\}$ with a small critical
exponent of $2+\sqrt{2}/2$. This represents the first progress on an open
problem of Vesti from 2017.
