Jan Draisma's chapter "Notherianity up to symmetry" in the book Combinatorial Algebraic Geometry (Springer LNM 2108) presents various finiteness theorems that are based on Kruskal's tree theorem (or actually the special case known as Higman's lemma).

The rest of the book also contains some potential examples, although there's some risk of getting tangled up in debates about where the combinatorics ends and the algebra or geometry begins.

Jan Draisma's chapter "Noetherianity up to symmetry" in the book Combinatorial Algebraic Geometry (Springer LNM 2108) presents various finiteness theorems that are based on Kruskal's tree theorem (or actually the special case known as Higman's lemma).

The rest of the book also contains some potential examples, although there's some risk of getting tangled up in debates about where the combinatorics ends and the algebra or geometry begins.