Does the lexicon of Ancient Greek contain words believed to be of Sumerian origin? If so, can some estimate of their number be provided?
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They were mostly borrowed via Akkadian, and into other major classical languages of the Eastern Mediterranean beside Ancient Greek - Aramaic, Armenian, Persian, Hebrew...
English cane would seem to share such an etymology.
Another wave of ultimately Sumerian words came to Southeastern Europe later via Ottoman Turkish, eg kireç. They are not found in literary Greek, but they are found in neighbouring South Slavic and Albanian, so surely they have been spoken in some Greek dialects.
When I remember right, the word βοῦς "cow" is traced back to Sumerian (it is believed to be a Sumerian loan into Proto-Indogermanic, the reconstructed form is *gʷṓws).
However, such beasts are rare and I doubt that there are many more of them, because Ancient Greek and Sumerian weren't in direct contact (temporarily and geographically).
EDIT: Here's another candidate: κύανος ‘dark blue glaze; enamel’, from Hittite kuwannan- ‘copper ore; azurite’ (ultimately from Sumerian kù-an).