This was so much easier than counting up the manuscripts of Vergil, which I don't seem to be anywhere close to finished doing.
Actually, Marielle de Franchis counted them up for me, in Chapter 1, "Livian Manuscript Tradition," of the Blackwell Companion to Livy, which was published in 2015, and a copy of which arrived for me via inter-library loan today.
Franchis mentions on p 5 "all the manuscripts of the First Decade (about 200) available today." 1 manuscript of a fragment of the Second Decade was found at Oxyrhynchus. On p 9, Franchis writes that "More than 170 manuscripts that transmitted the Third Decade between the fifth and the fifteenth cantury are still extant." On p 14, she tells us that the Forth Decade "has survived in about 100 manuscripts." There is 1 manuscript of the Fifth Decade containing books 41-45, and 1 containing a fragment of the Tenth Decade.
200 + 1 +170 + 100 + 1 + 1 = 473. The number is more likely to rise than to fall. By how much? I don't know.
I have admitted on this blog that I hope that many more missing parts of Livy's text will be discovered, and that I am aware that such hopes often make people chuckle who are much more learned on the subject of Livy than I. How much more learned? Well, for example, I have read Professor Michael Reeve's article "The Vetus Carnotensis of Livy Unmasked," in Studies in Latin Literature and its Tradition in Honour of C. O. Brink, ed Diggle, Hall & Jocelyn (1989), which Reeve wrote in my native language, English, read it several times, with the greatest interest, and I still am very far from comprehending its content.
So understand that my opinions on such matters, when they are not supported by citations of professionals, are decidedly amateur. My opinion that study of 6th-century Europe made lead to great discoveries of currently-missing parts of Livy's text? Amateur. I wouldn't be at all surprised if it would make the experts chuckle.
I think that they would be somewhat less inclined to chuckle (It doesn't hurt my feelings when they chuckle. Really, it doesn't) when I say that the number of manuscripts of Livy will rise from about 473, although the manuscripts added to the list will mostly (Here they may chuckle again, because I said "mostly" in stead of "all." It's okay) contain text currently known.
Faithful readers of this blog may have noticed that I've written a lot about the transmission of Livy's text, and almost nothing about the text itself. They may be thinking, "Heck, Steve -- what's so great about Livy anyhow?!" I may eventually write some answers to that question. I really do think that Livy is great: a wonderful writer who tells exciting stories, and occasionally underrated as an historian -- but even those who have called him worthless as an historian have agreed that he gives you a great read.