The canonical example of a war of annihilation is Hitler's war on the Eastern Front. Adolf Hitler, March 30, 1941, speech before the generals of the Wehrmacht:

''' Fight two worldviews against each other. Devastating verdict on Bolshevism is equal to antisocial criminality. It is a FIGHT FOR ANNIHILATION. If we do not take it that way, we will beat the enemy, but in 30 years the Communist enemy will be facing us again. We are not waging war to preserve the enemy. ... Fight against Russia: destruction of Bolshevik commissioners and Communist INTELLIGENCE. ... The fight will be very different from the fight in the West. In the east, hardness is mild for the future. The leaders must sacrifice their concerns.''' [2]

That war was also supposed to end in a couple - three months but it lasted 4 years. A formal, whitewashed definition of Vernichtungskrieg was given in [1]:

'''War Of Annihilation is a radical form of warfare in which "all psycho-physical limits" are abolished which permits the decimation or even total destraction of a population. The state organization of the enemy is to be destroyed. Another characteristic of a war of annihilation is the dehumanization and rejection of negotiations with the enemy. The legitimacy of the opponent is negated and the opponent is demoted to status of a “total enemy,” with whom there can be no negotiations. “Total victory” of one's own "Volk, Krieg und Politik” (people, war and politics) is the purpose and successful outcome (“triumph”) of a war annihilation.''' [1]

[1] Jan Philipp Reemtsma: Die Idee des Vernichtungskrieges. Clausewitz – Ludendorff – Hitler, pp. 386–398, Zitat p. 397.

[2] Franz Halder: Kriegstagebuch. Tägliche Aufzeichnungen des Chefs des Generalstabes des Heeres 1939–1942, Bd. 2: Von der geplanten Landung in England bis zum Beginn des Ostfeldzuges. Kohlhammer Verlag, Stuttgart 1963, S. 335 ff.; Christian Hartmann: Halder. Generalstabschef Hitlers 1938–1942. Schoeningh, Paderborn 1991, S. 241 f.