Barkskin = base 16 AC.
Bracers of Defense = +2 AC
Shield of Faith = +2 AC
Mage Armor = +3 AC
18 DEX = +4 AC
Ring of Protection = +1 AC
Stand next to a level 6 battle cleric with Herald of Battle = +1 AC
Have Shield and Counterspell slotted.
The result is a base 29 AC mage with the option to Shield up to 34 AC (yes, I have done this). Arrows and melee attacks are almost useless. Single target or area spells that affect the mage can be Counterspelled. Note that Barkskin and Mage Armor have long enough duration to be cast ahead of battle and can be cast from scrolls (Mage Armor scroll is a common recipe, too) and Shield of Faith is a bonus action so the first Paladin or Cleric who goes can cast it without using up an action.
Mage Armor shouldn't work like that with Barkskin. If it does, it's a bug. Mage armor sets your AC to 13 + Dex, and Barkskin sets your AC to 16 + Dex; therefore, the interaction between the two should be that Barkskin takes precedence over Mage Armor and provides 16 + all of the other bonuses.
That said, you've actually missed one other source of AC boost. If you pick up Shield proficiency (from one of the backgrounds or as a feat), you can wield a shield and enchant it with +3, for a total of another +5 AC. I believe in that case that Bracers of Defense would no longer function, meaning it would only be a +3 total. Maybe Ring of Protection as well, for a net +2.
Regardless, AC stacking is indeed easily achievable with magic items. It probably ought to be looked at from a balance standpoint to adjust the cost of these items to prevent quite so easily achievable near invulnerabilities.
Edit: There is no way to get shield proficiency through either feats or backgrounds currently in the game. I also confirmed with a quick built character, and Mage Armor and Barkskin both function as intended (i.e. Barkskin's buff overrides Mage Armors and only applies the 16 + Dex base AC it should).
Barkskin only raises your AC to 16 -- if it is lower than 16 -- and Mage Armor sets your AC to 13+DEX. With a DEX of 18 (+4) and Mage Armor your AC is 17 so Barkskin not only has no effect but is wasting your Concentration.
Not to mention that both Barkskin and Shield of Faith are concentration spells. So you would need your Battle Domain Cleric to cast SoF, when he/ she should be concentration on Bless or Flaming Sphere or Spirit Guardians, depending on their caster level and available spell slots.
there's much less expensive and micro-intensive way. Arcanist Sellsword upgraded to Iron Legion and given the nice dwarven plate with spellbook on shoulderstrap and cloak of displacement. All attacks prioritise other people and on the rare occasion they don't, you have AC 20 + whatever minor trinkets and optional longsword of Elai if you got the ingredients