How many non-biblical books are mentioned in the Bible and what are they?

The Bible is clear that some of its authors made use of outside sources. Luke is very general, and doesn't name his source documents (Luke 1:1-3); however, comparison between him, Matthew and Mark demonstrate that he clearly made use of Mark, as did Matthew, for that matter. Other points of comparison between Matthew and Luke have lead scholars to the conclusion that they both made use of another source document, thus far undiscovered, which is called, for convenience, Q (from the German word quelle, meaning source.)

Jude quotes from two Apocryphal books, The Book of Enoch in vs. 14 and The Assumption of Moses in vs. 9. Paul quotes from several Greek writers. In Acts 17:28 we see him using Aratus; in 1 Corinthians 15:8 he quotes from Menander and in Titus 1:12 he quotes from Epimenides.

In the Old Testament, the following non biblical books are mentioned:

1. The Book of Jashar (Joshua 10:13)

2. The Annals of Solomon (1 Kings 11:41)

3. The Book of the Kings of Judah and Israel (2 Chronicles 16:11, 25:26, 28:26, 32:32)

4. The Book of the Kings of Israel and Judah (2 Chron. 27:7; 35:27, 36:8)

5. The Book of the Kings of Israel (2 Chron. 20:34)

6. The Book of the Kings of the Kings (2 Chron. 24:27).

It is possible that these (3-6) may be four variant forms of the same title; it is also possible that they may be references to the books of Samuel-Kings.

7. The Book of the Kings of Israel (1 Chron. 9:1)

8. The Words of the Kings of Israel (2 Chron. 33:18)

9. The Annotations on the Book of the Kings (2 Chron. 24:27)

10. The Words of Samuel the Man of Vision and the Words of Nathan the Prophet and the Words of Gad the Seer (1 Chron. 29:29). May be one work; may refer to the books of Judges and Samuel.

11. The Words of Nathan the Prophet (2 Chron. 9:29; cf. 1 Kings 11:41-53)

12. The Prophesy of Ahijah the Shilonite (2 Chron. 9:29; cf. 1 Kings 11:29ff; 14:2ff, etc.)

13. The Visions of Jedo the Seer (2 Chron. 9:29; cf. 1 Kings 13)

14. The Words of Shemaiah the Prophet (2 Chron. 12:15; cf. 1 Kings 12:22ff)

15. "Shemaiah wrote" (1 Chron. 24:6)

16. The Records of Shemaiah the Prophet and Iddo the Seer that deal with genealogies (2 Chron. 12:15)

17. The Annals of Jehu the son of Hanani, which are recorded in the book of the Kings of Israel (2 Chron. 20:34; cf. 1 Kings 16:1, 7, 12)

18. "The rest of the acts of Uzziah, first and last, did Isaiah the Prophet, the son of Amoz, write" (2 Chron. 26:22; cf. Isaiah 1 and 6)

19. "The Vision of Isaiah....in the Book of the Kings of Judah and Israel" (2 Chron. 32:32; cf. 2 Kings 18-20; Isaiah 36-39)

20. References to "Lamentations" and "Jeremiah" (2 Chron. 35:25)

21. The Annotations of the Prophet Iddo (2 Chron. 13:22)

22. Liturgical writings of David and Solomon (2 Chron. 35:4; cf. Ezra 3:10)

23. Commandments of David and Gad and Nathan (2 Chron. 29:25)

24. The Commandment of David and Asaph and Heman and Jeduthun (2 Chron. 35:15)

25. Chronicles of King David (1 Chron. 27:24)

26. The Last Words of David (1 Chron. 23:27)

27. Letter of Cyrus (Ezra 1:1-4; cf. 2 Chron. 36:22-23)

28. Letter of Rehum and Shimshai (Ezra 4:6-16)

29. Letter of Artaxerxes (Ezra 4:17-22)

30. Letter of Tattenai (Ezra 5:6-17)

31 Memorandum from Babylonian Archives (Ezra 6:1-5)

32. Letter of Darius (Ezra 6:1-12)

33. Letter of Artaxerxes (Ezra 7:11-26)

34. Letter of Sanballat (Nehemiah 6:5-7)

35. The Book of the Wars of the Lord (Numbers 21:14)

Unsurprisingly, the corpus of Hebrew literature consisted of much more than what we call the Old Testament. The ancient Israelites were no different than the people of today who produce reams of literature, history, songs, and poetry. But not everything they produced was scripture. It is peculiar that anyone would be surprised by this.

Most of the literature, poetry, and historical annals mentioned in the OT have not, unfortunately, come down to us today. There is an extensive body of Jewish literature dating from about 400 BC through about 200 BC that is remarkable and interesting; parts are what are called the Apocrypha, which the Roman Catholic church has accepted on a par with the rest of the OT, though only since the Council of Trent in 1546. This was in reaction to Protestantism's explicit rejection of them as Scripture. Until then, they had been considered useful for teaching, but were not held in quite as high regard as the rest of scripture. Protestantism and Judaism have been universal in rejecting them as scripture.