I’m looking for the number of “kinds” of pieces, not the aggregate number of all pieces produced. A breakout of Duplo, Technic and other “specialties” would be helpful as well.
Bricklink will be a source of the data to answer this question.
At the time of this answer Bricklink has 59610 items in its part catalog. However, this list contains such categories like "Sticker Sheet" (3422 items), "Stickers Over Assembly" (250 items), "Special assembly" (38 items) you wish to exclude, since they are not really individual parts. This leaves us with 55900 parts. It is worth mentioning this number includes "decorated elements", which are printed or stickered version of the parts. Most of these decorated elements exist in non-decorated form, however there are exceptions.
Out of these 55900 elements DUPLO has 4299 entries.
Now Technic is a bit more complicated. If we take just the elements in Technic categories there will be 2298 parts. Some Bionicle and Hero Factory elements could also be considered as Technic, however they are listed under their respective categories instead.
Minifigure categories account for 17758 elements.
Noticeable categories include Bionicle (595 elements), Belville (158), Clickits (372), Modulex (1172) and Scala (465).
Some other categories are small enough to contain all elements within single category like Fabuland (223 elements), Galidor (117), Hero Factory (253) and Znap (21).