One interesting set of example is Salem numbers. These are real algebraic integer all of whose conjugates has absolute value less than or equal to one and at least one of the conjugates lies on the unit circle.
So you have a scenario of, one which flew over the cuckoo's nest, at least one in the edge and the rest are all in the nest (caged ?) .
Also if all its conjugates lies on the unit circle then it has to be a root of unity, its known as Kronecker Theorem.