First branch of the facial nerve is:

(a) Greater petrosal nerve(Ref. Scott Brown, 6th ed., 1385)The greater superficial petrosal nerve arises from the geniculate ganglion at the first genu. This is the first branch.The 2nd branch is the nerve to the stapedius given at the pyramid.The 3rd is the chorda tympani given 4-6mm before the exit of facial nerve from stylomastoid foramen.Lesser superficial petrosal nerve is not a branch of facial. It arises from tympanic plexus and supplies the Otic ganglion. Otic ganglion is for the parasympathetic supply to the parotid. Secretomotor fibres to the parotid arise from glossopharyngeal nerve, reach the otic ganglion through the lesser superficial petrosal nerve via the tympanic plexus and then reach the parotid through the auriculotemporal nerve arising from the otic ganglion.