I have a lv 5 Lizardfolk Barbarian with an AC of 42. Let me explain.

I got two 18's when rolling for ability scores and put them in DEX and CON. Lizardfolks race adds a +2 to your CON score so I have a 20 in CON at lv1.

When I reached lv4 I got the ability score improvement (ASI) so I put the +2 from that in DEX to get a 20 in DEX. 

Lizardfolks Natural Armor states while wearing no armor, you AC is 13 + DEX modifier which is a +5, giving me an 18 but lizardfolk can use shields with this trait and I had bought a shield so my Natural Armor AC became 20.

Barbarians Unarmored Defence states while wearing no armor, you AC is 10 + DEX and CON modifiers which was a +5 in DEX and a +5 in CON so that is 20 but barbarians can also use shields with this trait. 

So Lizardfolk comes in with a 20 and the Barbarian has a 22 so that's how I got a 42 for my AC. On top of this I carry around a boulder so I can have 3 quarters cover from that

What I want to know is, well, is this legal. My DM approved it but is there any book or something like that that states what the max AC is allowed. Also is their any way i can increase This? 

I know there are plus 3 shields, shields of faith but is there any other way to increace the AC on the Lizardfolk Barbarian I call Tank?

As I understand it, this is not even close to legal. I don't know where the rule is in the books or SRD, but you can only use one AC formula at a time. Bonuses from magic items, spells and whatnot stack on top of that base formula. So you could take the natural armor or the unarmored defense, but you can't add them together. Similarly, if you took a dip into Monk, you would not be able to add the wisdom bonus in. 

I think a more reasonable homebrew, if you and your DM wanted it, would be to take the lizardman's 13 and put the the DEX and CON on top of that, for a total of 23 + 2 for a shield = 25. Still sky high, and still not RAW.

IMHO, even this isn't necessary. The Barbarian in my campaign runs around with a greatsword and a loincloth and enemies struggle to put the hurt on him if he's raging. 

Out of curiosity, has the character ever been hit by an attack roll?

Not yet so bring on the Tarrasques and ancient dragons lol

Yeah, agree with FilthyHippy that you don't add the AC's together. In the same way that wearing leather armour and plate armour doesn't give you 11 + dex + 18!

So yeah, your choice is to either have an AC of 13+dex or an AC of 10+dex+con. Plus shield, if you're using it! So either 18 or 20, so that's a no brainer. 22 with shield. That's pretty sweet for not wearing armour!

Honestly, I'm not even sure it an AC of 42 is even possible in the game with every magical item stacked!

Quote from FilthyHippy >>

As I understand it, this is not even close to legal. I don't know where the rule is in the books or SRD, but you can only use one AC formula at a time.

It's in the section for Armor Class under characters:

Some spells and class features give you a different way to calculate your AC. If you have multiple features that give you different ways to calculate your AC, you choose which one to use.

Basically if you have more than one source of AC (including worn armour), then you have to choose one (and only one) to use, they do not stack in any way. It's actually technically possible to reduce your AC by wearing armour as a result of this (if you have better unarmored AC).

I don't think there's really any need to try and combine them; Lizardfolk do have other bonuses and plenty of races have redundant features depending what class they take, e.g- elf weapon proficiencies don't do anything extra if you go a Fighter.

Yeah that's not how AC is calculated, you don't add methods togheter, you pick one. There is a Sage Advice on this if you want to read more https://dnd.wizards.com/articles/sage-advice/sage-advice-compendium

How do you calculate a creature’s Armor Class (AC)? Chapter 1 of the Player’s Handbook (p. 14) describes how to determine AC, yet AC calculations generate questions frequently. That fact isn’t too surprising, given the number of ways the game gives you to change your AC! Here are some ways to calculate your base AC:

Unarmored: 10 + your Dexterity modifier.

Armored: Use the AC entry for the armor you’re wearing (see PH, 145). For example, in leather armor, you calculate your AC as 11 + your Dexterity modifier, and in chain mail, your AC is simply 16.

Unarmored Defense (Barbarian): 10 + your Dexterity modifier + your Constitution modifier.

Unarmored Defense (Monk): 10 + your Dexterity modifier + your Wisdom modifier.

Draconic Resilience (Sorcerer): 13 + your Dexterity modifier. Natural Armor: 10 + your Dexterity modifier + your natural armor bonus.

This is a calculation method typically used only by monsters and NPCs, although it is also relevant to a druid or another character who assumes a form that has natural armor. These methods—along with any others that give you a formula for calculating your AC—are mutually exclusive; you can benefit from only one at a time. If you have access to more than one, you pick which one to use. For example, if you’re a sorcerer/monk, you can use either Unarmored Defense or Draconic Resilience, not both. Similarly, a druid/ barbarian who transforms into a beast form that has natural armor can use either the beast’s natural armor or Unarmored Defense (you aren’t considered to be wearing armor when you use natural armor) What about a shield? A shield increases your AC by 2 while you use it. For example, if you’re unarmored and use a shield, your AC is 12 + your Dexterity modifier. Keep in mind that some AC calculations, such as a monk’s Unarmored Defense, prohibit the use of a shield. Once you have your base AC, it can be temporarily modified by situational bonuses and penalties. For instance, having half cover gives you a +2 bonus to your AC, and three-quarters cover gives a +5 bonus. Spells sometimes modify AC as well. Shield of faith, for example, grants a target a +2 bonus to AC until the spell ends. Magic items can also enhance your AC. Here are a few examples: +1 chain mail gives you an AC of 17, a ring of protection gives you a +1 bonus to AC no matter what you’re wearing, and bracers of defense grant you a +2 bonus to AC if you’re not wearing armor or using a shield.

Sorry, no. AC doesn´t stack. You have to choose the higher one. When your DM approved that, they should read the rules again, very carefully! And besides, a boulder of that size would weigh more than 1500kg, so too much to carry even for a barbarian. 

Hmm not sure about max AC but anything over 30 pretty much requires most enemies rolling a crit to hit you.  Off the top of my head go 2 levels Paladin for Defense Fighting Style, go Warforged for another +1 AC, +3 Plate and +3 Shield gets you to 28 add Shield of Faith and you hit 30.  Throw in a Ring and Cloak of Protection and you are at 32.  Grab 15 levels of Sword Bard and you can roll a d12 with Defensive Flourish, add in a Defender weapon and that's another 3, learn the Shield spell through Magical Secrets for another potential 5 and you can hit a decent AC of 40 + 1d12 so that's potentially up to a max of 52 AC.

That's just off the top of my head I'm sure I missed lots of stuff.  But like I said anything over 30 will pretty much require crits for most enemies to hit you.