What is the highest Armor Class achievable for a Sorcerer with the Sorcerous Origin of Dragon Bloodline?
Player's Handbook Chapter 6 variant rules included. (Multiclassing, Feats, etc.)
Total with equipment, feats, abilities:
Total including spell casting:
If it's a melee attack you can make use of Defensive Duelist, otherwise you can cast Shield instead. Both are reaction actions so you can only utilize one.
21 without magic. Higher with it
That's pretty much the universal answer to this question (with the barbarian being the potential exception). There are several ways to go about about this.
Pretty much the go to here is to MC something that gets you heavy armor and shields. Life or War cleric is the go to. You get access to Plate right away (as soon as you can afford it) and you can have an AC of 20 no problem. Another option is a class that gives medium and shields (Pally, Fighter, Ranger) and take a feat. Lastly you can go something that gives you light armor and shields, and have a Dex of 20.
Adding a MC to Ranger, Paladin or Fighter will let you boost this an additional 1 thanks the Defense Fighting style. This gets you to 21.
This will get you to 20. To get to 21, you'll need a magic set of plate. +1 armor is currently the highest level magic item published and it is listed as rare. Obviously the DMG will likely have higher level armor which will obviously push this higher. If I recall correctly it is rumored to cap out at +3 which would make the max possible AC 23 (It's unknown at this time if magic shields will stack with magic armor, the only magic shield published is not a +x item).
Additional items may push it further. Orvir notes several that have been published: Bracers of Defense, Staff of Defense and the Ring of protection. That could push you as high as 26 and a temporary bonus such as the shield spell he mentions could push you to 31 for a round.
However, if you don't want to MC, you can take the feat tree to get it (lightly, moderately, heavily), but at that point your ability scores will be kind of lousy, you'd likely be better of MCing.
A shield counts as armor so it won't work with the Draconic Sorcy's class feature, the max you can get there is 18 (unless you have a magic ability increase beyond the 20 cap for Dex, something we don't know exists at this point).