HistoryPorn

Allegedly, it happened around 50 B.C.

Cocaine or a Michael Jackson thing?

Hah.

Just in case, though, I'll clarify: what I'd heard was that, when the German army was in Egypt in WWII, some German soldiers used the nose for target practice and pulverized it. No aliens required.

Edit: I'm remembering the story wrong: the target practice thing is attributed to Napoleon's troops using the nose as target practice for cannon. It'd unsubstantiated in either case; it turns out no one alive really knows.

Strangely the recent documentation doesn't mention that.

As the article is lacking some information, here is a translstion of a part of the German Wikipedia article:

The Arab historian and physician Abd al-Latif al-Baghdadi (1161-1231) from Baghdad described the Great Sphinx and its magnificent nose in the 13th century. In the Middle Ages, the sphinx was still worshipped as a god by some sections of the population, while devout Muslims abhorred this cult. In Arab times, the Sphinx was given the name أبو الهول / Abū l-Haul, which means "father of terror". In one of his books, the Arab historian Al-Maqrīzī (1364-1442) reports that the devout sheikh of a Cairo Sufi monastery, Mohammed Saim el-Dar (Muhammad Şā'im ad-Dahr, English: "Someone who fasts all the time"), was a fanatical iconoclast who cut off the nose of the sphinx in 1378 and was then killed by the angry crowd.

The Danish artist Frederick Ludewick Norden (1708-1742) produced engravings of various Egyptian buildings in 1738 on the orders of King Christian VI. Among them was one with the buried Sphinx (Tête colossale du Sphinx), which also shows the head without a nose (published in French in 1755). The rumour that either Napoleon Bonaparte's soldiers or those of the Ottoman Empire destroyed the nose during artillery exercises has thus been proven false. Napoleon was an enthusiast of Egypt, describing the country as the "cradle of the sciences and arts of all mankind" (l'Égypte - le berceau de la science et des arts de toute l'humanité). The scientists who came to the country with him also drew the Sphinx without a nose.

TLDR: As the Great Sphinx still has had its nose around 1200 A.D. and was already noseless in 1738, its nose must have been destroyed in the meantime, supposedly by some furious sheikh in 1378.