The Hales-Jewett Theorem works in a context that is more abstract than that of the semi-group 
$\mathbb N$.  This is what makes it more applicable than earlier theorems.
This paper by Sabine Koppelberg points out some easy implications as well as a more general form of the Hales Jewett Theorem:
[The Hales-Jewett theorem via retractions. 
Proceedings of the 18th Summer Conference on Topology and its Applications. 
Topology Proc. 28 (2004), no. 2, 595–601]


I think some of the developments around Hindman's theorem, speaking about semi-groups more abstract (i.e., with fewer relations) than $(\mathbb N,+)$, for example the finite subsets of $\omega$ with union, are motivated by the abstractness of Hales Jewett.  Furstenberg and 
Katznelson used topological dynamics to prove a density version of the Hales Jewett Theorem.
Blass gave a proof of the Hales Jewett Theorem using ultrafilters.   The main result in
Koppelberg's paper mentioned above is also proved using algebra in the Stone-Czech compactification of a semi-group.
Here the Hales Jewett Theorem and variants seem to be interesting test questions for a method of proof.