How many species did Carl Linnaeus classify?
More than 13,000.

Plants: >9,000 names.
In Systema Naturae 10th edition, commonly taken as the starting point of modern taxonomy, Linnaeus is reported to have published around 6,000 plant names (I haven't counted, but Müller-Wille gives 5,900 and Stearn says "almost 6,000". The Wikipedia figure of 7,700 may come from a different edition of Systema Naturae).

However, that's just SN10. Luckily, a wonderful source has compiled the names from all of Linnaeus's work:
The Linnaean Plant Name Typification Project of the Natural History Museum says that Linnaeus published more than 9,000 valid plant names in his life (names that are still valid under current nomenclatural conventions), and they have many of them in a searchable database with references to where Linnaeus published them.

Animals: >4,200 names.
For SN10, different authors give 4,236 or 4,378 animal names. Stearn says "nearly 4,400", so perhaps he too was unsure. The total number Linnaues described in his life is probably higher, as he did write separate zoological publications like Fauna Svevica, but I couldn't find a source like the project bringing together all of his animal names.




 Müller-Wille S. 2006. Linnaeus' herbarium cabinet: a piece of furniture and its function. Endeavour 30: 60–64.
Stearn WT. 1959. The Background of Linnaeus's Contributions to the Nomenclature and Methods of Systematic Biology.)