In Hellenic religions few animals are linked to so many gods as the snake is. There is, really, no singular god or goddess that the snake enjoys sole attachment to, nor even any specific kind of domain other than perhaps a vague connection to chthonic gods (such as the Erinyes) and the earth. But in which case, you would find, that the majority of the Olympians, if one tried to fit all serpent imagery into this box, would have some sort of chthonic function….which in some cases one may struggle a bit more to find than in others. Actually, though chthonic imagery is credible, it seems more likely that the serpent has a greater association with divinity,or immortality in a general sense and the general ability of the serpent to be able to move between the upperworld and netherworld seemlessly (as it both burrows into the earth and sheds its skin) without death.
Zeus has a form of a serpent which is generally honored predominantly in his domestic worship for instance. This is usually associated with his chthonic movement, particularly in the case where in the form of a snake that he made his way into a cave where Demeter had hidden her daughter and there from their union she conceived Zagreus.This form of Zeus is also often identified with Agathos Daimon…but then perhaps Hera in such a case would be in her pairing with Zeus as an Agatha Tyche. Hera, while not usually associated with serpent in icons, is called the mother of Typhon, a being with a multitude of serpetine heads, in the Homeric Hymn to Apollon. She also seems to be associated with guardian dragons such as Delphinia,to whom she gave Typhon to nurse, and her sacred apple tree among the Hesperides is also guarded by a dragon.
Athena’s armor is draped in serpents and she has her connections to the serpent-haired Medusa. The serpents in this case may have something to do with her formidable, and rather lethal, nature as well her assistance of heroes to their own divinity (such as the slaying of Medusa by Perseus). Meanwhile, Ares is a father of dragons himself….or at least the father of the dragon which guarded the golden fleece sought by Jason and his Argonauts. His association with serpents seems to be of a similar nature to that of his mother, an association with violent guardian dragons which (like Delphinia and the Hesperide dragon) become part of the interplay between a hero,or heroic deity, and the accomplishment of his task which brings greatness.
Hermes, meanwhile has serpents on his wand, a gift from his brother Apollon in trade, that carries about with him as he, like serpents, moves between worlds. This may have some connection with a possible earlier role of Apollon, in connection with the movement of the sun (the sun’s chariot described by Euripedes in his Medea as being drawn by dragons), that descended to netherworld, but is one that is adopted by Hermes. Apollon, meanwhile has several serpents himself. His arrows are not only poetically likened to serpents, but he is the destroyer of the great dragoness Delphinia who often is represented alive along side Apollon in many icons. He,like the sun, has also been depicted with a dragon-drawn chariot. In many respects Apollon is the weilder of power symbolized by the presence of serpents and his control of this comes through the tale of Delphinia and their dual identification together that results after his grief and libations to the dragoness. The dragon chariot also has some link symbolically playing with harnassing the serpentine power.
Of course Apollon is not the only deity connected to serpents in this manner, as Demeter is also said to have had a dragon-drawn chariot which was used the sew wheat seed all over the earth according to myth. But this is rather natural anyway for Demeter to be so linked to harnassing the powers of immortality as the mother of the mysteries. This seems far more profound than a direct association of a chthonic role. For if the serpent symbolizes instead immortality, then the association of of serpents with gods shows more of how the divine interacts in the cosmos, and upon humanity. Zagreus, who is the spark within humanity, is conceived by the great serpent Zeus upon his daughter Persephone (born of the dragon-riding Demeter with Zeus). Other gods’ relationships with serpents plays out more in contest of growth for humanity specifically, coming to Apollon, the god of the boundaries, who is linked to immortal serpent that probably has a great deal to do with images of Apollon and Hermes weighing the souls of the dead heroes. And then there are serpents of Hermes that move between all the worlds…Hermes who carries off the dead and who receives the offspring of the gods. Meanwhile Dionysos’ maenads, carry serpents in images of their ecstatic union with the god. This is appropriate as wine, and Dionysos, acts as a kind of conduit between humanity and the gods…the undiluted wine associated with madness, even as the contact with the gods could potential bring a temporary ecstatic madness. Thus the ecstatic maenads, reveling in the embrace of the divine, are clad in serpents.
In my view this makes the symbol of the serpent much grander and far more complex.