Answering a question of Richomme, Currie and Rampersad proved that 7/3 is the infimum of the real numbers α > 2 such that there exists an infinite binary word that avoids powers but is highly 2-repetitive, i.e., contains arbitrarily large squares beginning at every position. In this paper, we prove similar statements about β-repetitive words, for some other β's, over the binary and the ternary alphabets.