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 This question is mostly about the modern times because I'm not sure if we can give a satisfactory definition for outdated political systems.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
In the USSR the head of state was collective: the Presidium of Supreme Council. The state was usually represented and laws signed by the Chairman of the Presidium, but he actually did not have any powers: if he would refuse to sign a law for instance, it would be signed by his deputy.