The Father/Son relationship seems to be paramount in Christianity, which seems odd when taken in the context of the gods of the Roman cities, the Greek pantheon, and the influence of other pluralistic theologies.
The need for a female relationship among Christians seems to have elevated Mary to near-divine status, but the movement failed to induct her to the circle of the divine, e.g. like Jesus Christ or the Holy Ghost.
Was there some kind of competition among religions to reduce the number of deity, or a political influence on Christianity that froze its fledgling pantheon at 3?
In my view, the answer is ‘Jewish Monotheism’.
Although its true that Christianity developed ina context shaped by Graeco-Roman polytheism, Christianity developed out of Judaism. The Judaism of the post-exilic period was, I would characterise, ‘fiercely monotheistic’. It had a very clear conception of one, sole, supreme God, “besides whom there is no other”.
The term ‘Son of God’ in the gospels, on my reading, draws primarily from Old Testament usage as a title affirming Jesus as the messianic King of Israel, rather than an ontological title trying to depict his relationship with God the Father. However, I do think the NT furnishes plenty of material that makes it reasonably to conclude, as say Bauckham does, that earliest Christians treated Jesus as God, and as no other god than the one God of the Jewish faith. This, of course, required some innovative thinking – how do you incorporate a second figure into the identity of a single and sole deity? How is that not polytheism? One thing they were clear on though was that it wasn’t polytheism and that was the wrong ‘model’ to apply. It takes early Christians a good 300 years more to figure out how they want to express that model, but it’s a model that conserves basic monotheism while creating a unique way of relating the oneness of God and the differentiation of Father, Son, and Spirit.
So there wasn’t really an ‘impetus’ to expand the number of ‘gods’ because on this view the starting point was a singular god besides which there could not be other gods. There certainly were groups that took to applying other models of how the Father and Son were gods, to include other spiritual beings as a kind of pantheon, but they both fell outside the mainstream tradition of Christianity as it developed and were largely unsuccessful in the long run. I would suggest that some forms of Gnosticism take that line.
There was, to my knowledge, no “kind of competition ..to reduce the number of deity”. Having a pantheon of 3 is actually very inconvenient (I don’t think ‘pantheon of three’ is a very accurate way to describe the Christian understanding of the Trinity). It lacks the philosophical simplicity and attractiveness of having but 1, and it lacks the range and diversity of being polytheistic. Neither would I suggest there was a political influence. Rather, I suggest the main impulse is within the religious texts and traditions of Christianity itself and its Jewish origins.