Let $A$ be a finite set. Let $A^k$ be the set of words in the alphabet $A$ of length $k$ and $A^*$ be the set of infinite words. I was looking for an element $a = \lbrace a_n \rbrace_{n \in \mathbb{N}}$ of $A^*$ so that, $\exists k_0 \in \mathbb{N}$ so that $\forall k>k_0$, any finite subsequence (of length $k \cdot m$) of $a$ is different from the sequences obtained by repeating some word at least $k$ times.

In other words, I was looking for an infinite sequence so that all subwords are not element of the diagonal in $X^k$ where $X= A^n$ and $n$ runs over all integers, $k$ runs over all integers larger than some $k_0$.

There is an [I suppose] absurdly complicated way to answer this by looking at a geodesic ray in the Burnside groups (this requires $|A| \geq 4$ and gives a $k_0$ around 665/2). Curiosity pushes me to ask

$\mathbf{Question:}$ How to produce (elementarily) such a sequence?