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\begin{abstract}
The sequence starts with $a(1) =1$; to extend it one writes the sequence
so far as $XY^k$, where $X$ and $Y$ are strings of integers, $Y$ is
nonempty and $k$ is as large as possible: then the next term is $k$.
The sequence begins 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3,
2, $\ldots$ A $4$ appears for the first time at position 220, but a $5$
does not appear until about position $10^{10^{23}}$.  
The main result of the paper is a proof that the sequence is unbounded.
We also present results from extensive numerical investigations
of the sequence and of certain derived sequences, culminating
with a heuristic argument that $t$
(for $t=5,6, \ldots$) appears for the first time at about position 
$2\uparrow (2\uparrow (3\uparrow (4\uparrow (5\uparrow \ldots 
\uparrow ({(t-2)}\uparrow {(t-1)})))))$,
where $\uparrow$ denotes exponentiation.
The final section discusses generalizations.
\end{abstract}

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