When was Jesus anointed and who did the anointing? I know that "Christ" or "Messiah" means "anointed or anointed one" and that Jesus commonly has the title of "Christ."  One thing I do not recall is an account of Him being anointed.  To that end, what is the biblical basis for Jesus' title?
Jesus of Nazareth was not anointed with the anointing oil which was typically used for the priests of the Levitical priesthood.1 Rather, he was anointed with the Holy Spirit, “the oil of gladness.”
In Psa. 45:6–7, it is written,

6 O’ God, Your throne is forever and ever; the scepter of Your kingdom is a scepter of righteousness. 7 You loved righteousness and hated wickedness. Therefore O’ God, Your God anointed You with the oil of gladness more than Your companions.

This event was documented in all four of the gospels.2 In addition, in Acts 10:38, it is written,

38 How God anointed Jesus of Nazareth with the Holy Ghost and with power: who went about doing good, and healing all that were oppressed of the devil; for God was with him. KJV, 1769

Augustine wrote,3

Who here, however slow he may be, does not recognize Christ whom we preach, and in whom we believe, when hears [that he is] God, whose throne is for ever and ever, and [that he is] anointed by God, as God indeed anoints, not with a visible, but with a spiritual and even intelligible chrism (chrismate)? For who is so ignorant in this religion, or so deaf to its far and wide-spread fame, that he has not known that Christ (Christum) is named from chrism (chrismate), this is, from anointing?
Quis non hic Christum, quem praedicamus et in quem credimus, quamlibet sit tardus, agnoscat, cum audiat Deum, cuius sedes est in saecula saeculorum, et unctum a Deo, utique sicut unguit Deus, non visibili, sed spiritali atque intellegibili chrismate? Quis enim tam rudis est in hac religione vel tam surdus adversus eius famam longe lateque diffusam, ut Christum a chrismate, hoc est ab unctione appellatum esse non noverit?

It is certainly true that Christians are also anointed by God the Father with the Holy Spirit who dwells in them.4 The difference is that the Lord Jesus Christ does not possess the Holy Spirit “from a measure.”5 The Greek word μέτρου, the genitive declension of the noun μέτρον, means a measuring instrument. God the Father did not restrict the quantity of the Holy Spirit given to His Son. Rather, “all the fullness of deity” dwells in the Lord Jesus Christ.6

References
Sancti Aurelii Augustini Episcopi. De Civitate Dei, Libri XXII. Vol. 2. Lipsiae: Teubneri, 1877.
Footnotes
1 cp. Exo. 29:7
2 Matt. 3:13–17; Mark 1:9–11; Luke 3:21–22; John 1:32–3; also, Luke 4:18 cp. Isa. 61:1
3 De Civitate Dei (“On the City of God”), Book 17, Ch. 16, p. 238
4 1 John 2:27
5 John 3:34: «ἐκ μέτρου»
6 Col. 1:19, 2:9