Awkward as it may sound, "joying a great joy greatly" is pretty close to the Greek of those biblical passages describing people's response to the coming of the Lord (see Matthew 2:10). Of course, not everyone is so eager for the coming of the Lord, but those who earnestly seek him are filled with a glorious and unspeakable joy (1 Peter 1:8).
Why such joy at the birth of a baby? Yes, we're all happy about the birth of any baby. But this baby was different, filling not only his parents with unspeakable and indescribable joy, but also even strangers with the kind of joy which is otherwise never, ever experienced. What was it about this baby which brought such joy?
The birth of this baby Jesus was a demonstration of God's great love and commitment to us—let us rejoice! The manger scene was a fulfillment of promises made long ago that God would not leave us in our fallen state—let us rejoice! The angel's announcement signaled that the time had come for a new beginning which will culminate in the new heavens and the new earth—let us rejoice!
This news eclipses even the joy of happy faces opening presents around the Christmas tree. In fact, without the coming of Christ, we'd be left to live in die in the worst sort of misery. So, for this reason, we "joy a great joy greatly."