Has there been a state without an office satisfying the definition of a head of state? This question is mostly about the modern times because I'm not sure if we can give a satisfactory definition for outdated political systems. Please feel free to comment on this, but still the main focus of this question is on the modern world.
Switzerland lacks a single head of state, though the members of the 7-head government take it in turns to represent the country as such abroad during the 7-year government term. See also Wikipedia on Politics of Switzerland.
While the U.S. was under the Articles of Confederation, the President of The United States in Congress Assembled was not a head of state. Instead, the role was to ensure impartiality of the congress and enforcement of rules. Articles of Confederation, Presidents of the Congress