Many see John's Prologue arranged using a literary device1
in which the first two verses are a single thought:
In the beginning was the Word, and the Word was with God, and the Word was God. This One was in the beginning with God. (John 1:1-2 DLNT)
ἐν ἀρχῇ ἦν ὁ λόγος καὶ ὁ λόγος ἦν πρὸς τὸν θεόν καὶ θεὸς ἦν ὁ λόγος οὗτος ἦν ἐν ἀρχῇ πρὸς τὸν θεόν
There are four statements, three linked by a καὶ followed by a fourth without καὶ. Both καὶ's are translated as the connecting conjunction "and." When understood as such it seems like the meaning is unaffected if they were left out:
In the beginning was the Word. The Word was with God. The Word was God. This One was in the beginning with God.
ἐν ἀρχῇ ἦν ὁ λόγος ὁ λόγος ἦν πρὸς τὸν θεόν θεὸς ἦν ὁ λόγος οὗτος ἦν ἐν ἀρχῇ πρὸς τὸν θεόν
This lack of additional meaning suggests they were not intended to be connecting; rather the writer had some other purpose which would add to the meaning to the passage. For example, the BDAG states καὶ may be used to introduce a result that comes from what precedes and would be translated "and then" or "and so" or it may be used to explain what goes before it and would be translated as "that is" or "namely:"2
In the beginning was the Word, and then/and so the Word was with God, and then/and so the Word was God. This One was in the beginning with God.
In the beginning was the Word, that is/namely the Word was with God, that is/namely the Word was God. This One was in the beginning with God.
Is one of the other uses of καὶ a better understanding of the use in the beginning verses of John's Prologue?
1. Two literary structures of the Prologue (1:1-18) recognize vv. 1-2 is a single unit. Marie-Émile Boismard, Le Prologue de St John identified a chiastic structure; Marc François Lacan, Le Prologue de saint Jean, identified three movements made up of three tracks, or waves.
2. Fredrick William Danker, A Greek-English Lexicon of the New Testament and Other Early Christian Literature, The University Chicago Press, 2000, p. 495