I challenge all of you to make a liberal democracy that bases its political legitimacy around the theory of divine right.
EDIT without any religious organization having any role in government.
Modern Iran does not really fit the "liberal" bit, but does otherwise fit the bill to some extent, depending how you regard the role of the "clergy" (well, "academia" would be a better word) in its government. There is an Islamic Republican understanding of politics that could easily lead that way, based upon the concept that the Muslim community as a whole cannot err, and thus a democracy would always follow the divinely guided path. The idea is very old and deeply rooted in Islam, but never got large scale institutional implementation before Modernity* and has been only imperfectly applied in most recent times. Iran doesn't exactly count in this regard as the inerrant community is a Sunni idea, while Iranian Shi'i Islamic republicanism is based on the vacancy of the Imam's office; the "divine right" there falls on the scholars who are caretakers for the absent Imam and (theoretically) have no divine right by themselves. There is a large opposition movement in Morocco that wants exactly that (a religiously inspired democracy based on the divine right of the community, as opposed to the king) but, while they have a significant following, they are nowhere near power or any likelyhood of getting it. However, I think that Islam may offer a more fertile ground for democratic divine right than Christianity.
* Khariji groups in North Africa and Ismailis in the Gulf created small "republics" on this basis in the Middle Ages, some of these relatively long-lived. It was pretty marginal, doctrinally and geographically, and the largest and stablest of these things always leaned toward re-establishing a sort of dynasticism albeit limited. Also, by no means they were liberal democracies, of course. The governing community was strictly sectarian and never truly egalitarian within, with a few notables being paramount.