A string attractor of a string $T[1..|T|]$ is a set of positions $Γ$ of $T$ such that any nonempty substring $w$ of $T$ has an occurrence that crosses a position in $Γ$, i.e., there is a position $i$ such that $w = T[i..i+|w|-1]$ and the intersection $[i,i+|w|-1]\cap Γ$ is nonempty. The size of the smallest string attractor of Fibonacci words and period-doubling words of length at least $2$ is known to be $2$. The size of the smallest string attractor of Thue--Morse words of length at least $25$ is known to be $4$. We completely characterize the set of all smallest string attractors of Fibonacci words, period-doubling words, and Thue--Morse words. For Fibonacci words, we show a recursive formula describing the $2^{n-4} + 2^{\lceil n/2 \rceil - 2}$ distinct position pairs that are the smallest string attractors of the $n$th Fibonacci word for $n \geq 7$. For period-doubling words, we show that the $n$th period-doubling word has exactly two smallest string attractors for every $n\geq 2$, and give them explicitly. For Thue--Morse words, we show a formula describing the 32 sets of four positions that are the smallest string attractors of the $n$th Thue--Morse word for $n\geq 6$. Our results show that strings with the same smallest attractor size can have a drastically different number of distinct smallest attractors.

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