Under unusual circumstances, it is possible to get an AC of 49

Highest permanent AC: 31
Highest combat-ready AC: 35
Highest the-stars-aligned AC: 49

Permanent AC with Magic Armor, no other Magic Items: 27

There are actually several ways to achieve this.

Barbarian 20
22: Unarmored Defense = 10 + 5 (20 DEX) + 7 (24 CON)
+5: +3 Shield

NOTE: Barbarian 20 precludes several options that require levels in other classes

Heavy Armor
21: +3 Plate Mail
+1: Defense Fighting Style (Fighter 1 or Ranger 2 or Paladin 2)
+5: +3 Shield
Medium Armor
18: +3 Half-Plate
+3: 16 DEX or better and Medium Armor Master Feat
+1: Defense Fighting Style (Fighter 1 or Ranger 2 or Paladin 2)
+5: +3 Shield

Things that don’t work:

Light Armor: 26
15: +3 Studded Leather
+5: 20 DEX
+1: Defense Fighting Style (Fighter 1 or Ranger 2 or Paladin 2)
+5: +3 Shield
Robe of the Archmagi: 25
15: Robe of the Archmagi (Wizard, Sorcerer, or Warlock 1)
+5: 20 DEX
+5: +3 Shield
Monk: 20
20: Unarmored Defense = 10 + 5 (20 DEX) + 5 (20 WIS)
+0: Precludes shield

Adding Magic Items with Ongoing Effects (Except ones that permanently raise ability scores): 31

+2: Staff of Power (Wizard, Sorcerer, or Warlock 1)
+2: Choose two out of Ring of Protection, Cloak of Protection, Ioun Stone – Protection

NOTE: The Defender, because of the way it works, is considered (and rejected) as a Situational AC Boost.
Adding Combat Buffs – Consumable Magic Items and Spells: 35

+2: Potion of Speed (effect of Haste spell, no Concentration)
+2: Shield of Faith (Cleric 1 or Paladin 2, Concentration)

Adding Situational AC Boosts: 49

+5: 3/4 Cover
+4: Multiattack Defense (Ranger 7, triggered by getting hit, active for rest of attacking creature’s turn)
+5: Shield (Wizard 1, reaction to being hit, active until your next turn)

A note on the Defender: The Defender and Staff of Power are both weapons that can provide a significant AC boost. I think most characters with the choice would choose the Staff of Power. In hand it is always on and also has a +2 attack or spell DC bonus. The Defender is not in effect during any round you do not attack, including a surprise round or a round in which you Disengage or Dodge, and going max AC leaves no bonus for the attack roll. Importantly for this question, hiding behind 3/4 cover pretty much means you are not attacking, and therefore cannot shift +3 to AC. If the Defender replaced the Staff of Power and chose not to hide behind 3/4 cover, the highest AC would be 45.

Bonus Questions: What’s the lowest level at which I could achieve this?

Because this doesn’t rely on crazy stats, it can be achieved at a fairly low level. We have a choice between a Heavy and a Medium Armor build. Using the Medium Armor build obviates the need for Heavy Armor Proficiency (which would require a feat or 1st level in Fighter or Paladin).

Minimum Stats (easily achievable at 1st level using the default array with any PH race):

DEX 16
INT 13 (Wizard multiclass)
WIS 13 (Ranger and Cleric multiclass)

Feat:

Medium Armor Master – taken at Ranger 4, or at 1st level as a Variant Human

Minimum class level requirements:

Ranger 7 / Wizard 1 / Cleric 1

Important Caveats

I have assumed that ability scores max out at 20 unless a class feature says otherwise. If abilities go above 20, where do they stop? I have chosen to disregard two possible ways in which characters can get ability scores over 20:

Even without magic items, Epic Boons can be used to continue to improve abilities to 30 (DMG 230), while the characters remain “20th level” characters. Given enough time, assume all 30s.
There are several magic items that give permanent ability boosts. The only ones worth considering here are the Tomes and Manuals which give +2 to the relevant ability but also increase the ability maximum by 2. Although, once used, they become blank for a century, there is nothing in these manuals that suggest they are unique or that they can’t benefit a character multiple times. How high can your ability scores go? Per RAW and RAI, the answer to Can I increase an ability score above 30? is NO.

In either case, we no longer need Barbarian 20 for CON 24. Barbarian 1 gets access to Unarmored Defense, time/magic gets us DEX 30, CON 30, and multiclassing is possible. Highest the-stars-aligned AC becomes 57. I think this violates the spirit of the question however.