  We study the lexicographically least infinite $a/b$-power-free word on the
alphabet of non-negative integers. Frequently this word is a fixed point of a
uniform morphism, or closely related to one. For example, the lexicographically
least $7/4$-power-free word is a fixed point of a $50847$-uniform morphism. We
identify the structure of the lexicographically least $a/b$-power-free word for
three infinite families of rationals $a/b$ as well many "sporadic" rationals
that do not seem to belong to general families. To accomplish this, we develop
an automated procedure for proving $a/b$-power-freeness for morphisms of a
certain form, both for explicit and symbolic rational numbers $a/b$. Finally,
we establish a connection to words on a finite alphabet. Namely, the
lexicographically least $27/23$-power-free word is in fact a word on the finite
alphabet $\{0, 1, 2\}$, and its sequence of letters is $353$-automatic.
