You want to find rings having some properties but not having other properties? Go there: Database of Ring Theory! A great repository of rings, their properties, and more ring theory stuff.
This page provides counterexamples on Clifford algebras, spinors, spin groups and the exterior algebra. You’ll find a list of counterexamples references at the end of the article.
Look at pi-Base for a gold mine of topological examples.
Want to know more about counterexample definition? Look at Wikipedia.