A parliamentary system? In other words is your question about a "Head of State" not elected by popular vote? Oh hell there is a bunch of them friend.
Canada, Turkey, Austria, Germany, Italy, Israel, Australia, United Kingdom, Netherlands, Belgium, Ireland, and India. Many of these countries have a President (e.g. Ireland, Germany, and Israel) but that person is a mere figurehead head of state like the monarch in a constitutional monarchy. Some of the parliamentary systems do elect a president by popular vote but others have the parliament elect the president.
There are currently four major hybrid countries (where it is unclear who is more powerful the President or the Prime Minister) -- Lebanon, France, Russia, and Pakistan. In two of the four (France and Russia), the President is elected through popular vote. In Pakistan, the President is elected by an electoral college composed of the national parliament and the provincial assemblies. In Lebanon, the President is elected by Parliament.