  We begin a systematic study of the relations between subword complexity of
infinite words and their power avoidance. Among other things, we show that
  -- the Thue-Morse word has the minimum possible subword complexity over all
overlap-free binary words and all $(\frac 73)$-power-free binary words, but not
over all $(\frac 73)^+$-power-free binary words;
  -- the twisted Thue-Morse word has the maximum possible subword complexity
over all overlap-free binary words, but no word has the maximum subword
complexity over all $(\frac 73)$-power-free binary words;
  -- if some word attains the minimum possible subword complexity over all
square-free ternary words, then one such word is the ternary Thue word;
  -- the recently constructed 1-2-bonacci word has the minimum possible subword
complexity over all \textit{symmetric} square-free ternary words.
