Alright. Let's work this out, since my friends and I have as well.
High Elf Barbarian 1 (Let's assume, for all intents and purposes, you started with maxed stats, 18s in all (DEX to 20 and 19 INT due to High Elf).
Level 1 has you sitting at a 10+5+4 Nekkid AC of 19. Throw in a +3 Shield, and you're at 24. I have a good reason for going Elf instead of where many do.
Next, go Wizard for 2 levels to get the Blade Dancer Arcane Tradition to add your INT Modifier too, as it isn't an "Unarmored Defense" equivalent. So you can get another +4 to that AC, but you lose the Shield at this point. Make sure one spell you do take is "Mage Armor" to increase your Base AC to a 13.
So for Level 3 (Barbarian 1 / Wizard 2) you'd have 13(Mage Armor)+5(Dex)+4(Con)+4(Int) = 26 AC with NO MAGIC ITEMS.
For the next 5 levels, we'd work on leveling those two classes to 4 so you get the Ability Score Increase. Improving Constitution or Intelligence each time since an ASI from this ability can't improve it beyond 20. This would raise it up two more points by 8th level to a 28 AC.
At this point, additional levels don't REALLY matter since you can't improve Base Scores through ASI, and since wearing armor is going to lose you 3 or more points from the AC score, you're not going to want to mess that up. And wearing a shield, will get you no net gain (+3 shield give +5 AC, which loses the Blade Singing bonus which is your INT modifier, also a +5 at this point).
From here on out, any and all improvements will be gained via Magic Items or through playing with RAW.
One way, as has been said, is to get the Ability Score improving books. As a High Elf, you're likely to "live well over 700 years" which, for sake of argument, we'll say you started adventuring as an elf at the child-like age of 50. We'll say you stole these books from the library of your starting city, not realizing what they were. The ones that would benefit you most as a Barbarian 4 / Wizard 16 (Since for some of this, I need access to 7th+ level spells) would be: Manual of Bodily Health (Constitution), Manual of Quickness of Action, and the Tome of Clear Thought. Assume you read each of these which increase the scores by 2. The first time reading these gets you a +3 to that already large AC, a 31. Then, let's attune our first magic item, Bracers of Defense, this raises it to a 33 since we're not wearing armor, or using a shield, throw in a Ring of Protection, and an Ioun Stone of Protection, and we're at a 35 (+1 AC from both). Next, we'll assume we use a full "Deck of Many Things" and get REALLY lucky with our draws and get the Star and boost our INT up and raise it to a 36. We'll also assume we get the Moon and max our "Wish" number (These come into play later) at 3.
Alright, as it stands we're at a whopping 36 AC. Let's progress the clock until you're 150 years old. You've just gained another +3 and you're now at 39. Lather, Rinse, Repeat. 250: 42 AC, 350: 45 AC, 450: 48 AC, 550: 51 AC, 650: 54 AC, 750: 57 AC. By this time, you're getting up in years, despite it saying "Well Over 700" we're unsure how old they actually intent to allow you to get. So we'll assume you won't make it to 850... naturally...
Now we're messing with the RAW. By this point you've probably got a well established family and are well on your way to becoming a legend. You know that you can tell your family to leave your three books alone, and they would.
So you go ahead, and use a spell slot and cast "Sequester" on yourself with the set rule to not be awakened for 100 years. You'll be left there, perfectly fine, unless someone finds and dispels the magic before the 100 years. You can use this method, basically indefinitely to raise it well beyond the 57 AC we're sitting at before that.
You could also use your ability from the "Wish" spells you technically gained through the Moon card to wish to be able to lower the age of your physical body to a point such as 50 (Where you started), but to retain all the improvements and mental advantages you had gained through your years of experience... Basically, this one works BEST if you are a lawyer and can actually prevent any and every loophole possible to keep your skills and scores where they were to get you that 57 AC as a 50 year old elf, and possibly start your career anew and start using the books again. Or use a new set that you could have made during your wait since you'd have had the, well over 6,000 days to make a new set (2,000 each since they're each 50,000 and divide that by 25 to make to determine the days required). From that point on, you could go through that process two more times without having to worry about needing to find a new way to get "Wish."
tl;dr End Tally is a 57 AC without hitting a ceiling for the age of a High Elf.