Challenge to Christian Belief: The Trinity

Why three persons? Why not four or sixteen persons? 

Wouldn't the greatest conceivable being have an infinite number of persons, as that would be the highest level of being?

Or maybe being one person is greatest, as having your being split among persons is to somehow be divided, or made of parts, or to lack perfect coordination or perfect unity?

Or maybe it's just incoherent to have multiple persons in one being, as each person is necessarily a being? After all, we say that the cat is not the mat that it lays on, because the cat and the mat are made up of different properties. We distinguish the two by their properties, and by seeing that the properties are different, we see that the cat is not the mat. Likewise, when we see that one person has one set of mental properties, and we see that another person has a distinct set of mental properties, we see that these two persons are separate. And just as the cat and the mat are separate beings, or objects, so too must separate persons be separate beings.

And if the Holy Spirit is a person on par with the Son and the Father, then why isn't the Holy Spirit seated at the right hand of the Son, or at the left hand of the Father? (Hebrews 1:3, Hebrews 12:2, 1 Peter 3:22, Acts 7:55, Acts 2:33, Matthew 22:44, Revelation 3:21.) We consistently see the Father and the Son paired, with the Holy Spirit missing, almost as if the Holy Spirit is something shared between the two, or represents the presence of one or the other, or of both, as felt on earth by believers despite the lack of a bodily presence.

Maybe Jesus plays a unique role in being seated next to the Father, because Jesus has a body and the Holy Spirit does not? But the Father does not have a body either, and yet is still depicted as on a throne.

Jesus commands that we baptise in the name of the Father, Son, and Holy Spirit (Matthew 28:19). But why couldn't that be a redundancy? Jesus says that we should love God with all our heart, soul, and mind (Matthew 22:37; Deuteronomy 6:5). So redundancy is a familiar feature in the commands of Jesus.

But confusingly, there are verses that treat the Holy Spirit as a who, a person separate from the Son or Father. Consider John 15:26 (NRSVUE):

"When the Advocate[f] comes, whom I will send to you from the Father, the Spirit of truth who comes from the Father, he will testify on my behalf."

(Curious: In John 14 the Father sends the Holy Spirit on behalf of the Son; in John 15 Jesus says that He sends the Holy Spirit.)