Since Kruskal's tree theorem is a generalization of Higman's lemma, and the famous TREE function is based on Kruskal's tree theorem, one could try to define an analogous "WORD" function.
WORD($n$) is the largest $m$ such that there is a sequence $W_1,\ldots,W_m$ of words over an $n$-ary alphabet where $W_i$ has at most $i$ letters, such that $W_i \le W_j$ (subsequence order) does not hold for any $i<j\le m$.
The lemma doesn't immediately prove that $m$ is bounded, but I think this does: otherwise by the pigeonhole principle infinitely many of these sequences start with some word $x$, of these infinitely many follow with $y$ etc, so we can have an infinite subsequence-free sequence $(x,y,\ldots)$, contradicting the lemma.
If I didn't mess up anything this is kind of an obvious thing to define, and yet I couldn't find anything about it. What can be said about WORD($n$)? How does it compare to TREE($n$) (besides "$\le$")?