Artifact Armor +1

You can upgrade your Artifact Armor by trading it along with various items and Ancient Beastcoins to Sagheera in Port Jeuno. If it's your first time talking to her click the option "Just wanted to chat" first to open up the menu of Artifact attires.
Sagheera requires one item that drops in Apollyon and one from Temenos; these are the same for the whole set. She also requires a synthesis item and 15-40 Ancient Beastcoins, depending on the piece. This is what she will say:

I came from the Near East to peddle my wares. I heard that there are simply hordes of gullib--err, that is, I have heard of the great adventurers that roam this land, and wish to be of service to them. If you happen to know of any seasoned adventurers, I ask that you bring them to me. Be sure to tell them of my fabulous good!
Wait a moment...
You are an adventurer, are you not? I hear your sort does all kinds of things--fighting with monsters, cooking, and collecting all sorts of little knickknacks...
Oh, I am truly sorry if I have offended you. I am from a land far to the east of here, and tales of adventurers are quite popular in my hometown.
Anyway, you adventurers are quite the rowdy lot, always playing outside and showing up looking as if you had wallowed in mud, correct?
Are you really so dirty?
[...]
Oh my, such confidence.
Could it be because you possess that remarkable armor I have heard about in rumors...artifact armor, I believe it is called?
I am not finished, so wait! Undoubtedly your artifact armor was once something to take pride in, but at present it is surely quite battered from your adventuring, not to mention that it has simply become old and could use a boost...correct?
If I were you, my dear friend, I would take your precious artifact armor and restore it! Rejuvenate it! Give it new life! Does that not sound like the proper course of action?
I know exactly what you need. You absolutely must bring the necessary ingredients, and I will send them with your artifact armor to the craftsmen of the Near East for a miniscule fee in ancient beastcoins!
There, I have contracts with the most skilled artisans, who will make your artifact armor shine like never before.
Of course, it shall be difficult for you to give up your beloved artifact armor, if but for a short while. However, please inform me if you work up the courage, and I will tell you the ingredients I require.

Trade her the Ancient Beastcoins first, then the rest of the items. Wait until the next Conquest update and then talk to her again to receive your AF+1.

The upgraded pieces are all Level 74. If you want to get the original AF back, you can restart the AF quests by talking to Vingijard.