How many species did Carl Linnaeus (senior) 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.
Indeed this is a bit of interesting history. Linnaeus was not a modest man, but he was also a prodigious contributor to biology. He made many editions of his two major works Species Plantarum (1753) and Systema Naturae published in 1759. Systema Naturae covered both plants and animals and had 12 editions, eventually with 3 volumes in several parts.
Linnaeus' sampling of species was not broad - he traveled to Lapland France, and England. His main experience was in botany and was able to look at samples in gardens which were a popular pursuit amongst wealthy hobbyists of the day and visited botanists in England and had samples from other continents. He had heard about chimpanzees, but it doesn't seem as if he ever saw one.
Wikipedia gives a final count of nearly 10,000 species including over 7000 plants, from the entries in his works. Linnaeus felt that there would be very few more to be discovered; he estimated 10,000 species of plants which turned out to be completely wrong. The Encyclopedia of Life has 1,316,775 entries today. This was probably perpetuated by the influence of Aristotelian thought, which persisted through Darwin's life, that species were only the result a higher reality creating animals attuned to a local climate and geography and so the same species would in the mountains of Scandinavia would be the same as the alpine regions of India say. But even Aristotelianism, which stood for over a millenium the 19th Century and Darwin's work, was probably more the result of confirmation bias than any real proof.