What is the maximum AC a PC can acquire without the use of magic items?
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You can't use a shield with the Bladesong, dropping you to 34+1d10+1d8; however, by holding a second weapon in your off hand you can take the Dual Wielder feat, adding another 1AC. The Fighter's Defense style doesn't work either (no armour), so you're down another point.
We can also get behind three quarters cover, adding an extra 5, bringing us to 39+1d10+1d8.
Finally, let's drop three levels of Bladesinger and five of Bard, replacing them with Hunter Ranger 7/Cleric 1. The loss of Haste is replaced with the Cleric's Shield of Faith; the Bard drops to a d8, but in exchange we can get Multiattack Defense, which gives us +4 to AC versus a creature who's already hit us.
Final total: 43+2d8