Talks using just two words, such that the same thing is never said three times in a row

Ah, a beautiful riddle. The answer is the Thue-Morse sequence. Defined as follows: Starting with $a(0)$, $a(n)$ is equal to $B$ if $n$ has an odd number of $1$s in his binary representation, and $I$ otherwise. The theorem (never saying the same thing thrice in a row) is a special case of "no overlaps", where overlap means $xAxAx$ where $x$ is a letter and $A$ is a (possibly empty) string of letters. The proof of "no overlaps in the Thue-Morse sequence" is in this paper: http://arxiv.org/abs/0706.0907. If there is some nonempty string of letters $A$ that's said thrice in a row, $AAA$, then $A=xB$ where $x$ is the first letter of $A$ and $B$ is the rest of the string ($B$ might be empty), then there is $AAA=xBxBxB$, but even $xBxBx$ doesn't appear in the sequence, thus $xBxBxB=AAA$ doesn't appear.