I don't think there is any right that is unbreakable. A person can choose to not have medical treatment even if it kills them, meaning the state is willing to allow someone's right to life to be infringed (by that person). A person's right to a medical treatment can come up against someone else's right to life (in the case of abortion). Right to free speech is balanced against immediate public panic (the "fire in a crowded theatre" thought experiment), and is also balanced against someone's right to a good name (i.e. libel and slander law). The right to own property is balanced against the state's right to tax you. The right to privacy is balanaced against your child's right to not be abused (meaning the state can look at what you're doing). etc. etc.
Can you succinctly name a right that never needs balancing against another?
