While writing about the immutability of God, I entirely intended to address this question: Does Jesus becoming man mean that God changed, that He became mutable? My word count limit, however, came a lot faster than I realized, so I have decided to write a follow-up article addressing this question.
Although my answer is a definitive no, some would indeed argue that God became mutable during Christ’s incarnation. The line of reasoning is typically associate in some way or another with the kenosis theory. Holders of this theory argue that Philippians 2:7, which uses the Greek word kenoó to say that Jesus emptied himself, refers to Jesus giving up, or at least severely limiting, His divinity in order to become man. If the kenosis theory was a reality, it would indeed negate the immutability of God because Jesus, as God the Son, would have ceased being fully God for thirty-ish years.
Fortunately, Philippians 2:7 does not mean that Jesus surrendered over His Godhood while taking up His manhood. Instead, the whole Christ hymn of Philippians 2:5-11 is celebrating that though Christ humbled Himself into becoming one of us and even dying upon the cross, the Father has subsequently exalted Him as Lord of all. Thus, as Sproul notes:
the emptying was an emptying of divine prerogatives. Jesus laid aside his privileges. He voluntarily humbled himself. His taking on human nature did not subtract anything from his divine nature, but cloaked and concealed his glorious and exalted divine nature. On many occasions Jesus willingly concealed his divine authority and power. Although he warned now and then that he might call upon heaven for a display of power (summoning legions of angels, for example), in his restraint, he fulfilled the role of the obedient servant to the bitter end… He allowed himself to be arrested, tried, scourged, mocked, and crucified by men whom he could have annihilated with a single glance. That is humility. Acting with less power than one has at his disposal when attacked by another displays an astonished level of grace. This is the Jesus who is our Lord. It is precisely his mind-set that Paul called us to imitate in Philippians 2.Sproul, Enjoying God, 118-119.
This is what makes the nature of Christ, as most notably declared in the Definition of Chalcedon, a supreme mystery alongside the nature of the Trinity. Just as there is only one God who is eternally existent in three Persons, so too is Jesus one person with two distinct natures. Such notions are beyond our finite grasp, yet the testament of Scripture clearly indicates that Jesus’ incarnation in no way changed His divinity.
According to His human nature, Jesus did indeed change (although never once committing sin). As Luke notes of Christ’s childhood, “Jesus increased in wisdom and in stature and in favor with God and man” (3:52). Yet His divine nature remained unchanged, immutable. The incarnation did not diminish, in any way, the deity of Christ, for “Jesus Christ is the same yesterday, today, and forever” (Hebrew 13:8). This is the great mystery and unspeakable wonder of the eternal and divine Word becoming flesh and dwelling among us (John 1:14).
 We see this particularly in His repeated use of God’s holy name, I Am, for Himself.