Spoiler alert: Do not read this article if your faith depends on the literal historicity of Bible accounts; it will only upset you. I have called myself an "Emmanuelical Christian," meaning that what's most important to me about the historical Jesus is the concept of God dwelling fully within a human life. That said, I both embrace the story we've just retold and refuse to argue about the details. The short answer to this title is "nothing," but nothing can stop it from being a great story.
The story is cobbled together from the Gospels of Matthew and Luke, thought to have been written within five years or so of each other, 85 to 90 years into the "Common Era" of restarting the calendar from the opening of that story, which happened only centuries later. Both of these sources drew on the Gospel of Mark, from 15 or 20 years before, and the source from as much as 50 years before that scholars know as Q, but they didn't get their birth narratives there. Neither Mark nor the fourth canonical Gospel, John (also about 90 CE), includes any birth narrative. Though authors such as the early Bart D. Ehrman tried to make the Gospels out as "four independent historical records," their inconsistencies just do not hold up to scrutiny as history is done today.
Going with what we have, however, here's where the different elements of our familiar story are "attested" in the Bible:
Jesus' DNA: Both Matthew (1:1-17) and Luke (3:23-38) provide genealogical background. Matthew starts 42 generations back, to Abraham, includes several women notable mostly for their questionable purity (the mother of King Solomon, by King David, is named only as "Uriah's wife"), and ends with "Joseph the husband of Mary, of whom Jesus was born." Luke gives a flashback from the start of Jesus' ministry, starting with Joseph and includes no women.
Annunciation to Mary: Only in Luke (1:26–38). He also has the exclusive report of her "Magnificat" response (verses 46–56) and the rest of her visit to Elizabeth, mother of John the Baptist.
Joseph's reaction: This is Matthew's exclusive, and focus for eight verses (1:18–25), ending with naming the newborn.
Bethlehem, shepherds, angels: The heart of the familiar story can be read only from Luke (2:1–21). Unfortunately, there's no evidence of any such Caesarian decree or Roman census of the empire, though taxation was based on censuses and had been since David's time. And Luke has no star, just an angel appearing to the shepherds. Luke continues for seven more verses with elements sometimes left out of the narrative, though they were vital to establishing Jesus' propriety, at least as an infant, before Jewish audiences of Luke's time. These include circumcision, naming, and taking him to the Temple in Jerusalem, where he is recognized by the prophets Zechariah and Anna. And afterward (vv. 39–40), the family returns to Nazareth, where "The child grew and became strong."
Magi: Back to Matthew (chapter 2). That's right, if we take the canonical gospels as historical records, the "Three Kings" and their star should never be in the same Nativity scene with the shepherds and angels. That also means the Magi's warning dream to go home by another way (verse 12), the "slaughter of the innocents" (2:16–18), and most significantly, the family's escape to Egypt (13–15) appear only in Matthew's gospel. Matthew's story does end in Nazareth, but not for some years, and not as a return, but as further hiding from authorities and fulfillment of prophecy (19–27).
I don't mean to try to take this story away from anyone. Part of me wishes I could still hear it myself in the coherent, glorious form that I did as a child. But it has actually become more important to me as two tellings of the basic grace of incarnation, each revealing something different about how two different early Christian communities understood God and grace. I may not set up my creche anymore, but I feel more authentically connected to my forbears in one faith that has spanned two thousand years, a global expansion of the known world, many languages, and unfathomable technologies of communication. I don't need to accept or reject any details of these tellings; the words are less important to me than the witness of those ancient believers and the relationship with them that I am privileged to gain from their words.