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:
The phrase “only-begotten” is not an accurate translation and should not be used in any of the nine passages. This phrase is derived from the Latin Vulgate (a translation of the Bible from about the 5th century which has been quite influential on other translations) and reflects certain theological debates about the person of Christ. While the language of the Word being born of God is present in the 2nd century, the most notable occurrence of this language is the creed from the Council of Nicaea in A.D. 325. This creed speaks of the Son of God as begotten of the Father, unique—that is, from the substance of the Father—God from God, light from light, true God from true God, begotten not made, of one substance with the Father. This creed was the result of the rejection of the heresy that the Son of God was the first created being. Ultimately the phrase “begotten not made” leads to what theologians call the doctrine of the eternal generation of the Son. (Elwell, W. A., & Beitzel, B. J. (1988). Baker encyclopedia of the Bible (1590). Grand Rapids, MI: Baker Book House.)
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:
And some assign this to the day of his incarnation, when he declared him to be his Son, and that he should be so called, as Luke 1:35; some to the day of his baptism, when he was again solemnly from heaven proclaimed so to be, Matt. 3:17; some to the day of his resurrection, when he was declared to be the Son of God with power, Rom. 1:4, and Acts 13:33; some to the day of his ascension, whereunto these words are applied. And all these interpretations are consistent, and reconcilable with each other, inasmuch as they are all means serving unto the same end, that of his resurrection from the dead being the most signal amongst them, and fixed on in particular by our apostle in his application of this testimony unto him, Acts 13:33. And in this sense alone the words have any appearance of respect unto David, as a type of Christ, seeing he was said, as it were, to be begotten of God when he raised him up, and established him in his rule and kingdom. (Owen, J. (1854). Vol. 20: An Exposition of the Epistle to the Hebrews, Volume 3 (W. H. Goold, Ed.). Works of John Owen (136–137). Edinburgh: Johnstone and Hunter.)
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.
What, then, is time? Philosophers use the word in two closely related senses; most of us find that one of these is sufficient for us. Time, say the philosophers, is the duration of that which changes; time, say the philosophers again and we with them, is the measurement of the changes of the universe. What is common to both statements is the relation of time to change. Where nothing changes, time has no possible meaning. Thus time and the universe started together. God is infinite and therefore changeless. He is “the Father of lights with whom there is no change or shadow of alteration” (James 1:17). He possesses the utter fullness of existence, so that nothing can go from Him, for He already possesses all. The universe He created is a changing universe. And because change belongs to it and not to God, time belongs to it and not to God. To repeat, time and the universe started together: time is the ticking of the universe.
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:
We, then, following the holy Fathers, all with one consent, teach people to confess one and the same Son, our Lord Jesus Christ, the same perfect in Godhead and also perfect in manhood; truly God and truly man, of a reasonable [rational] soul and body; consubstantial [co-essential] with the Father according to the Godhead, and consubstantial with us according to the Manhood; in all things like unto us, without sin; begotten before all ages of the Father according to the Godhead, and in these latter days, for us and for our salvation, born of the Virgin Mary, the Mother of God, according to the Manhood; one and the same Christ, Son, Lord, only begotten, to be acknowledged in two natures, inconfusedly, unchangeably, indivisibly, inseparably; (ἐν δύο φύσεσιν ἀσυγχύτως, ἀτρέπτως, ἀδιαιρέτως, ἀχωρίστως – in duabus naturis inconfuse, immutabiliter, indivise, inseparabiliter) the distinction of natures being by no means taken away by the union, but rather the property of each nature being preserved, and concurring in one Person (prosopon) and one Subsistence (hypostasis), not parted or divided into two persons, but one and the same Son, and only begotten God (μονογενῆ Θεόν), the Word, the Lord Jesus Christ; as the prophets from the beginning [have declared] concerning Him, and the Lord Jesus Christ Himself has taught us, and the Creed of the holy Fathers has handed down to us.
The Nicene Creed was formulated by the Early Church Fathers to authoritatively define the Christological absolutes that orthodox Christians hold:
I believe in one God, the Father almighty, maker of heaven and earth, of all things visible and invisible. I believe in one Lord Jesus Christ, the Only Begotten Son of God, born of the Father before all ages. God from God, Light from Light, true God from true God, begotten, not made, consubstantial with the Father; Through him all things were made. For us men and for our salvation he came down from heaven, and by the Holy Spirit was incarnate of the Virgin Mary, and became man.
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:
There may still remain one error clinging to our knowledge of the processions of the Persons in the Blessed Trinity because of our own immersion in time. As far as the statement of it goes, we are not likely to make the error of thinking that the Son is in some way less eternal than the Father, or the Holy Spirit in some way younger than the Father and the Son. We know that there is no succession in eternity, no change in God. God the Father did not first exist as a Person and then become a Father. God, by the very act of being God, generates his Son; God the Father and God the Son, by the very act of being God, spirate the Holy Spirit. As I say, there is not likely to be any error in our statement of this: the error will tend to cling to our idea in such a way that when we are looking directly at it, we do not see it, yet it is profoundly there: and, because time is so deeply woven into all our experience, our advance in the knowledge of God depends upon our deliberate effort to rid our mind of it.
According to the historical Church's creeds and councils, the only valid answer is, "Christ is eternally begotten only of the Father."
The Logos has two nativities begotten eternally from the Father before all ages and born of the Virgin for our salvation.
If anyone will not confess that the Word of God has two nativities, that which is before all ages from the Father, outside time and without a body, and secondly that nativity of these latter days when the Word of God came down from the heavens and was made flesh of holy and glorious Mary, mother of God and ever-virgin, and was born from her: let him be anathema.
Second Constantinople (553), Anathemas Against Three Chapters, Canon 2.
The first nativity refer to His begotten eternally from the Father while His second nativity is always referred to as born of the Virgin not begotten. Subtle difference but theologically significant. Because begotten refer to His eternal generation while born refer to His temporal conception by Theotokos. The same self Logos has two nativities He was begotten once in His divinity and born once in His humanity.
[T]his whole contemplation begins in the Scripture, because it begins with Jesus. Jesus is the Son of God. Well, how is he the Son of God? He’s the Son of God because he’s begotten of the Father, meaning he has no human begetter. He has no human father. His Father, literally, is God. And God who is his Father begets him before all ages, and then this very one, who is God’s Son, is born as a man, as a human being, from the Virgin Mary. And it’s interesting that, in Greek, the same verb, when it applies to a father, is “beget”; when it applies to a mother, it’s “born.” So we wouldn’t say that Jesus was “begotten” of Mary humanly; he was “born” of Mary humanly. But we would also not say that he was “born” of the Father; he was “begotten” of the Father. Those are important things to think about, and that’s why we’re talking about them now. We have to contemplate these things and get it straight, get it clear.
Fr. Thomas Hopko, The Holy Trinity.
The same Greek verb can be used in two distinguishable ways: one to designate the eternal relationship between the Father and His Son (μονογενής) and another to designate the temporal relationship between the Theotokos and her Son (γεννήθηκε).
Begotten of His Father before the worlds according to His Godhead; but in these last days for us men and for our salvation born of the Virgin Mary, the Mother of God according to His manhood.
Pope St. Leo the Great, Tome (448).