I'm not sure how he's accomplishing that with an 18th level single classed fighter- its probably multiclassed in some way. But there are a few other feats, like Heavy Armor Optimization and Greater Heavy Armor Optimization, that collectively provide +2 to AC and I believe -3 to armor check penalties.
Since you'd need an AC of at least 77 to pull off 'everything in the Monster Manual needs a 20 to hit' (rather than check the whole MM, I just looked at the Tarrasque and a Great Wyrm Red Dragon), I'm not sure how he pulls it off. I mean, just with the DMG I can figure up to AC 49 before feats, class abilities, and oddball magic items pretty quickly with by-the-book 18th level wealth.
Start as 20 dex small 18th level character, then improve your dex 4 times, to give a base AC of 18, and 440,000 gp. Buying a ring of protection +5, bracers of armor +8, gloves of dex +6, a heavy shield +5, a dusty rose ioun stone, an amulet of natural armor +5, and a manual of quickness of action +4 will get you an AC of 49 and 100,000 gp.
But that still means you need to find another 26 points of AC between oddball magic items that stack with all that, feats, and class abilities.