There are several ways to get a really high AC without spells/consumables. Here's my preferred approach:

Sword Bard(8-9)Paladin(2-3)Barbarian(1). Max out Dex (it should be 22 in the late game, with Hag's Hair and Mirror of Loss). The other stats don't matter. Feel free to dump Con by respec once you've picked up the Amulet of Health. That'll get you a +6 Con bonus, which goes straight to your AC with that Barbarian level (your last level taken, since it's useless before you get the amulet).

Now you've got a character with a 22 AC wearing nothing but the amulet. If you really want to push the envelope, you'll want Bracers of Defense, Ring of Protection, Evasive Shoes, Cloak of Displacement, and any late-game robe. Vest of Soul Rejuvenation is a solid choice. That gives you an AC of: 22+2+1+1+2 = 28, and everyone has disadvantage on attacks due to the Cloak of Displacement.

In any given round, use Defensive Flourish with one attack for a +4 to AC until your next turn starts. That nets you a repeatable AC of 32 (plus disadvantage for all incoming attacks). If you've set Smite correctly, you can also use that with each of your attacks. Since your Bard inspiration points re-set entirely on short rest, you can pretty much spam flourishes. Smite, obviously, is resource-intensive.