We’ve discussed semirings before. One interesting application is tropical geometry, which studies the analogue of algebraic varieties over the max-plus semiring (sometimes known as the tropical semiring).
Grigory Mikhalkin has posted a survey article on the subject, Tropical geometry and its applications, to arXiv. (The “applications” of the title are applications to ordinary algebraic geometry.)

Update. Commenter ansobol has compiled an online bibliography of recent works in tropical geometry . For pre-1996 works, there is another bibliography by Stephane Gaubert.