How many times was Jesus of Nazareth, whether pre-incarnate or incarnate, begotten?
Once - in two different ways depending on our definition of 'begotten'
There seems to be some confusion on the subject because "only begotten" a theological term does not mean "begotten" a biblical term.
But to answer your question, if thinking 'begotten' as in 'only begotten' it gains prominence in Council of Nicaea in A.D. 325 representing Christ’s single eternal generation:
Not to say that the theological term is not very biblical. When the word begotten is used with “son” it does mean God’s only Son, indicating God’s absolute delight towards him and his absolute uniqueness (Jn 3:16, 18; 1 Jn 4:9). The statement at the baptism and transfiguration of Jesus in the Synoptics 'This is my beloved son' also carries the same sense of God’s absolute delight towards him and his absolute uniqueness, but not principally a literal/spiritual 'birth'. Not 'principally' a 'birth' but by implication as Jesus is unique since he is also eternally begotten. If that does not make him unique I do not know what can. In fact the very notion of a son, and all its uses in scripture could be argued as having its ultimate meaning in dimly reflecting the glory of the Son. The concept of sonship finds its ultimate meaning in Christ.
To confuse matters more, in Hebrews Chapter 1:5 it reads ‘For to which of the angels did God ever say, “You are my Son, today I have begotten you”? Or again, “I will be to him a father, and he shall be to me a son”? this seems to imply a 'birth' or sorts, but not really. From this type of reference we are naturally tempted to mix the theological term (i.e. eternal generation) with the biblical prophecy first fulfilled in David (as a type) and in Christ as Messiah. Therefore, although “only begotten” is a theological term referring to a single eternal event, ‘today I have begotten you’ means something totally different in terms of a true generation, or 'birth'. This really refers to his 'birth' from the dead, or the first born.
John Owen explains the different views about 'begotten' very well here, as well as the correct view to take:
Conclusion: The biblical term “begotten” refers to the resurrection of Christ, which happened once. The theological term “only begotten” refers to the eternal generation of the Son, by an inconceivable communication of the essence and substance of the Godhead by the Father --- only happened in a continuous forever. God’s absolute delight towards Christ and his absolute uniqueness implied by the words 'only Son' brings the single eternal generation and the single generation from the dead together under the Sonship theme. Both his eternal generation and his resurrection make him absolutely unique and proven fully approved of by God. This is partly why Jesus always used the phrase 'the Son' as implying being one and only person generated from yet equal to his Father, for example, as in John 5:16-29.
In order to answer this question logically, we must address 3 theological concepts:
1 – Time
2 – The Incarnation of the 2nd person of the Trinity
3 – The eternal state of the procession of the Son from the Father
In light of Thomistic theology, first, let’s focus on time.
One of the best (if not the best) Augustinian/Thomistic theologians is Frank J. Sheed. His great work “Theology and Sanity” is a wonderful beginner’s guide to navigating St. Thomas Aquinas’ Summa Theologica.
If we say, “Jesus was begotten before the universe was created,” then we are stating something that has no meaning at all. Before is a word of time, and there could be no time before the universe because time began with the universe. To say “before the universe” means when there wasn't any “when”; which is to say that it doesn't mean anything at all. The same is said about the phrase, “how many times was (was is also a time word) Jesus begotten.”
The Miaphysite heresy - which holds that the human nature and pre-incarnate divine nature of Christ were united as one divine human nature from the point of the Incarnation onwards - was officially denounced at the Council of Chalcedon.
The Confession of Chalcedon provides a clear statement on the human and divine nature of Christ:
The Nicene Creed was formulated by the Early Church Fathers to authoritatively define the Christological absolutes that orthodox Christians hold:
According to this formula, Christ is "the only begotten Son of God." While being eternally "born and/or begotten from the Father before all ages," he was "by the Power of the Holy Spirit incarnate of the Virgin Mary, and became man." It is precisely because Jesus of Nazareth was eternally begotten/born (fully God and fully Man) before all ages, that the Blessed Virgin Mary can properly receive the coronation of being named "Theotokos," or Mother of God.
These are all mysterious truths that leave human finite intelligence in the dark. Thomistic theology wonderfully helps to shed light on the infinite mystery of the Father eternally begetting the Son.
Frank Sheed goes on to discuss processions in eternity:
According to the historical Church's creeds and councils, the only valid answer is, "Christ is eternally begotten only of the Father."