The symmetric ring of quotients \(Q_ s(R)\) is an important subring of the Martindale ring of quotients of the prime ring R. It was introduced by V. K. Kharchenko in his work on Galois theory. It was known that \(Q_ s\) need not be a closure operator, but in the example given \(Q_ s\) was a closure operator on \(Q_ s(R)\). This paper constructs a domain R whose full sequence of iterated symmetric rings of quotients does not stabilize.