Gordan's lemma is another application of "compact + discrete => finite", but it is one of the basic building blocks of the theory of toric varieties.

In some sense the theory of division by polynomials (i.e. the Gröbner basis theory) is a manifestation of compactness, e.g. Dickson's lemma can be stated and proved as an application of compactness, see e.g. the wikipedia entry. I guess this is an example of the principle mentioned in the answer of Michael Greinecker.