It depends on how you quantify 'most war', and even then it's difficult to pin point. If by 'most war' you're referring to the scale of conflict, then Nazi Germany, whose actions led to the biggest conflict the world has seen so far.
If, however, you're referring to frequency and duration of war, then we must definitely look at the Ancient World, even before Rome. Historical records point to the fact that empires in the Middle East, such as the Akkadian and Assyrian, were known to conduct military campaigns almost every year. The Neo-Assyrian Empire, in fact, was famed for this, leading campaigns against Urartu (modern day Armenia and Turkey), Israel, Judah, Egypt, and Babylon. It's attempt to control the latter led to many years of almost continuous conflict.
You also have to take into account political entities whose entire existence was based on war. Alexander's Empire, the Mongol Empire, Napoleon's Empire and the Japanese Empire definitely, in one way or another, fit into this category.
One up and coming country in this field is the USA. Ever since the end of their isolationist policy, during the Second World War, the USA has been involved in an increasing number of conflicts which have led to an increasing number of deaths. This has somewhat intensified since the beginning of the War on Terror, with interventions (in one way or another) in Afghanistan, Iraq, Yemen, Somalia, Libya and now Syria. Their destabilization of the Middle East has also led to a large amount of conflict, the latest manifestation of which is the Islamic State.