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At least 41,684 words.
Why "at least"? Because it isn't always clear what Johnson considered to be a unique word.
The very first entry in the dictionary, for A, illustrates this problem. Johnson writes fourteen paragraphs of explanation for A, some of which describe different parts of speech and different meanings. Here are three such paragraphs:
Did he consider these to be different senses of the same word, or different words with the same spelling?
We could try to answer the "how many words" question based on capitalization and location: if a word is written in all capital letters, and if it is outdented from the column of text, then it's a headword that Johnson is defining.
Unfortunately, the Dictionary's formatting isn't consistent enough for us to be confident of this definition. For example, Johnson capitalizes hip-hop and gives it a separate definition, but the word is also numbered as though it's just one of the senses of To Hip.
By contrast, the entry for To Quinch says "This word seems to be the same with queech, winch, and queck"; of these three synonyms, queech does not have its own entry in the dictionary - in fact, this is the only place queech occurs in the entire Dictionary--but it isn't capitalized or outdented here. Should we count queech as a word that Johnson defined?
And then there are entries that list what are obviously forms of another word, such as kept. The word kept is the same word as To keep . . . isn't it? Johnson doesn't provide a separate definition for kept, possibly because he provides three separate entries for keep (and one of them, keep, v.a., lists 40 different senses). Should we subtract kept from the count, even though it is outdented and in all-caps?
Finally, Johnson often links multiple words to the same definition. For example, did he consider necerchief to be a unique word or just a different spelling of neckatee?
It's no wonder that different people have arrived at different answers to the question of "How many words did Johnson define in his dictionary?"
Given all of these conundrums, how did we arrive at our answer, "at least 41,684 words"?
We counted our entry files. For the infrastructure of the Johnson's Dictionary website, we created a separate computer file for every identifiable entry, defining entry as "the chunk of text headed by at least one outdented all-capitalized word."
All of the examples above are contained in single files: all fourteen paragraphs about A are contained in a single file, the words to hip and hip-hop are contained in a single file, the words necerchief and neckatee are in a single file, etc. Because we have 41,684 such entry files, we know Johnson defined at least 41,684 words.
If you decide to count the words yourself, let us know how many you find!
The first edition of Johnson’s Dictionary was issued in two substantial folio-sized volumes. Each volume contained about 1,150 pages, each page about 18 inches tall and 10 inches wide. The Dictionary defined nearly 43,000 words, plus it included three introductory essays: Preface, The History of the English Language, and A Grammar of the English Tongue. The whole thing weighed more than 20 pounds, making it similar in size and weight to a sewing machine, a large turkey, a stack of four 4x8” landscape pavers, a temporary “donut” spare tire.
Johnson himself declared the book "Vasta mole superbus" ("Proud in its great bulk")
The Latin poem on the title page is from Horace's Epistles, where Horace is addressing those who want to write poetry that will be considered great. Here is a translation by Andrew Wood (Edinburgh: William P. Nimmo, 1872):
Johnson’s Dictionary has remained in print in some form since it was first published in 1755. J.D. Fleeman’s majesterial reference work, A Bibliography of the Works of Samuel Johnson: Treating His Published Works from the Beginnings to 1984 (Clarendon Press, 2000), devotes 245 pages to listing various editions, re-issues, abridgements, and adaptations of Johnson’s Dictionary. These include 60 editions based on the original folio, three facsimile editions, 127 abridged editions, 47 adaptations and translations, and 316 “miniature” editions. Additional editions have been published since 1984, including a CD-ROM edition and two abridgments for popular audiences.
The miniature editions, in particular, represent at most 75% of such volumes, Fleeman notes. Although they have no authoritative relationship with the Dictionary proper—they generally contain lists of words explained by 1-2 synonyms—they illustrate Johnson’s popular reputation as a lexicographer.
Of the numerous editions of Johnson’s Dictionary, the 1st folio edition  has been the most accessible and has received the most scholarly attention. The 2nd folio (1755-56) edition arrived on the heels of the first, sold in affordable weekly numbers to appeal to a wider range of readers. Johnson revised the “Preface” to that edition, but probably not the wordlist. The 3rd edition (1765) was essentially a reprint of the 2nd; its few revisions were designed to help it function as a glossary to Johnson’s edition of Shakespeare, with which it was published to coincide. However, the 4th edition , the last edition published with Johnson’s participation, was extensively revised by the author, with new headwords, new senses, and new quotations (perhaps as many as 16,000 changes.) This edition was so thoroughly revised that any project of this kind would be incomplete without it.
For more about Johnson’s work on different editions, see Allen Reddick’s The Making of Johnson’s Dictionary 1746-1773 (Cambridge University Press, 1996).
Paul Luna describes the Dictionary’s typography in his essay "The typographic design of Johnson's Dictionary" (Anniversary Essays on Johnson's Dictionary, edited by Jack Lynch and Anne McDermott, Cambridge University Press, 2005):
"In type design, the faces cut by William Caslon in the 1720s and 1730s provided a systematic (though not wholly uniform) set of related roman, italic, and small-cap founts in a full range of sizes. [...] Johnson's printer, William Strahan, bought his types from the Scottish typefounder Alexander Wilson, who offered faces 'conformable to the London types,' in other words close in design to those of Caslon." (p. 179)
This question is difficult to answer exactly, but of the original printing of 2,000 copies of the 1st folio edition , probably fewer than 1,000 copies still exist.
A search of the WorldCat database found 384 libraries holding copies of the 1st folio edition . However, a single library can have more than one copy. For example, The Warren N. and Suzanne B. Cordell Collection of Dictionaries holds two copies of the 1st folio edition  (and hundreds of other editions https://library.indstate.edu/rbsc/cordell/johnson2.html ) Also, some libraries do not subscribe to WorldCat. Their copies, plus copies in private collections, are difficult to locate.