NPCs aren't limited like PCs are
Like Thomas Markov mentions in his answer, it is up to the DM to dictate monster statistics and you are perfectly within your right to edit the AC of your BBED to whatever figure makes the game enjoyable to you and your players.
What if I use RAW options?
If you were sticking to the options that are listed in the rulebooks, you can get a number of different answers depending on how your BBED wants to appear. To start with, the highest AC with a default ancient green dragon is 29. This is achieved by attuning to the following magic items (worn items resize to fit the wearer by default):
- staff of power
- two among ring of protection, cloak of protection, ioun stone: protection
... and the Blessing of Protection alternate reward. Then have minions cast warding bond and shield of faith on the dragon.
If the doors are open to artifacts, you could get quite a bit higher at an AC of 33 with the same Blessing of Protection and spell buffs, along with these magic items:
- staff of power
- Orrery of the Wanderer (with the +1 AC random property in addition to the +1 granted by default)
- Wand of Orcus (with the +1 AC random property in addition to the +3 granted by default)
If you chose to give the dragon class levels in accordance to the that option in the Dungeon Master's Guide, you could get as high as 50+ or 54+ with the artifacts. This is done with the following class distribution:
- Barbarian 1 --------------------- (for Unarmored Defense)
- Wizard (Bladesinging) 2 ------ (for Bladesong)
- Ranger (Hunter) 7 ------------- (for Multiattack Defense and the Martial Adept feat)
- Bard (College of Swords) 10 -- (for Defensive Flourish and Magical Secrets for haste)
This gives the following base AC calculation:
10 (base) +
10 (Dexterity after reading 9 manuals of quickness of action) +
10 (Constitution after reading 3 manuals of bodily health) +
10 (during Bladesong; Intelligence after reading 5 tomes of clear thought) +
2 (from the haste spell)
...with the potential for an additional 4 from Multiattack Defense, 1d6 from Evasive Footwork, and 1d10 from Defensive Flourish.
Then you just add the magic items and buffs from before to get the totals I mentioned.
Another important utility for AC is access to the shield spell. If you are using classes, this is gained from being a wizard, but otherwise the dragon can get access once per day through the following variant in the Monster Manual:
VARIANT: DRAGONS AS INNATE SPELLCASTERS
Dragons are innately magical creatures that can master a few spells as they age, using this variant.
A young or older dragon can innately cast a number of spells equal to its Charisma modifier. Each spell can be cast once per day...
...or through a magic item (which would replace the least beneficial item from before):
- a ring of spell storing gives 5 castings before it needs to be refilled by some minions
- non-obsidian spell gems (which can be refilled) or spell scrolls (which are consumed) can each grant 1 additional casting if the dragon knows the spell from the variant rule
- staff of defense gives 10 charges where casting shield as an action (meaning it hampers the dragon's damage output when it casts the spell since it can only attack through Legendary Actions) costs 2, and it takes an average of 1.833 dawns to recover all the charges.
The haste spell is another AC granting spell that a minion could cast (instead of the dragon itself like in the class levels case) on the dragon for +2 AC, but I would advise against it. If that minion ever loses concentration or dies, then the dragon would be affected by:
the target can't move or take actions until after its next turn, as a wave of lethargy sweeps over it
That being said, if this is a dynamic you would want to include in the fight, you can add 2 to the AC through this spell.
A Physical Shield
Another way to get a bit more out of the AC is to have the dragon equip a shield. This is a bit absurd in my opinion, but if you want to allow this, the dragon could get +5 AC from a +3 magic shield. The Basic Rules say after all:
When a nonhumanoid tries to wear an item, use your discretion as to whether the item functions as intended. A ring placed on a tentacle might work, but a creature with a snakelike tail instead of legs can't wear boots.
...so a DM could rule that a dragon can use a shield.