Was about to post the first 19 it found, and it got a 20 as I was writing it up:
Underlings Intrigue Base Baron Cellar Festival Library Masquerade Minion Nobles Pawn Steward Witch
Gambits Hinterlands Hinterlands Cartographer Crossroads Embassy Inn Jack of all Trades Mandarin Nomad Camp Oasis Oracle Tunnel
Instant Gratification Hinterlands Prosperity Bishop Expand Hoard Mint Watchtower Farmland Haggler Ill-Gotten Gains Noble Brigand Trader
Grim Parade Dark Ages Dark Ages Armory Band of Misfits Catacombs Cultist Forager Fortress Hunting Grounds Knights Market Square Procession
Chess With Death Dark Ages Dark Ages Bandit Camp Graverobber Junk Dealer Mystic Pillage Rats Sage Scavenger Storeroom Vagrant
Invasion Dark Ages Intrigue Diplomat Harem Swindler Torturer Upgrade Beggar Marauder Rogue Squire Urchin
Gentle Intro Adventures Adventures Amulet Distant Lands Dungeon Duplicate Giant Hireling Port Ranger Ratcatcher Treasure Trove Scouting Party
Expert Intro Adventures Adventures Caravan Guard Coin of the Realm Haunted Woods Lost City Magpie Peasant Raze Swamp Hag Transmogrify Wine Merchant Mission Plan
King of the Sea Empires Seaside Archive Farmers' Market Overlord Temple Wild Hunt Explorer Haven Native Village Pirate Ship Sea Hag Delve Fountain
Big Time Empires Prosperity Capital Gladiator/Fortune Patrician/Emporium Royal Blacksmith Villa Bank Forge Grand Market Loan Royal Seal Dominate Obelisk
Silver & Gold Base Base Bandit Bureaucrat Chapel Harbinger Laboratory Merchant Mine Moneylender Throne Room Vassal
Grand Scheme Intrigue Base Bridge Mill Mining Village Patrol Shanty Town Artisan Council Room Market Militia Workshop
Dusk Nocturne Nocturne Blessed Village Cobbler Den of Sin Faithful Hound Fool Monastery Night Watchman Shepherd Tormentor Tragic Hero
Midnight Nocturne Nocturne Conclave Crypt Cursed Village Devil's Workshop Druid Exorcist Leprechaun Pooka Raider Secret Cave
Collector Guilds/Cornucopia Seaside Fairgrounds Farming Village Fortune Teller Harvest Hunting Party Embargo Fishing Village Merchant Ship Navigator Smugglers
Collider Guilds/Cornucopia Seaside Menagerie Horn of Plenty Horse Traders Jester Tournament Lighthouse Salvager Treasure Map Treasury Warehouse
Illuminati Guilds/Cornucopia Alchemy Butcher Herald Masterpiece Merchant Guild Stonemason Apprentice Golem Philosopher's Stone Scrying Pool University
Tonics & Toxins Guilds/Cornucopia Alchemy Baker Candlestick Maker Doctor Plaza Soothsayer Alchemist Familiar Herbalist Transmute Vineyard
Overture Renaissance Renaissance Acting Troupe Experiment Flag Bearer Hideout Improve Inventor Lackeys Old Witch Seer Treasurer Fair
Prelude Renaissance Renaissance Border Guard Cargo Ship Ducat Mountain Village Priest Recruiter Sculptor Silk Merchant Swashbuckler Villain Citadel Star Chart
Makes good use of the new Guilds/Cornucopia sets, with Seaside paired with Cornucopia twice and Alchemy paired with Guilds twice. "Only" 10 single set kingdoms.
[edit]
Down to 9 single set kingdoms:
Underlings Intrigue Base Courtier Diplomat Minion Nobles Pawn Cellar Festival Library Sentry Vassal
Big Actions Prosperity Prosperity City Expand Grand Market King's Court Loan Mint Quarry Rabble Talisman Vault
The Duke's Ball Hinterlands Intrigue Conspirator Duke Harem Masquerade Upgrade Duchess Haggler Inn Noble Brigand Scheme
Travelers Hinterlands Seaside Cutpurse Island Lookout Merchant Ship Warehouse Cartographer Crossroads Farmland Silk Road Stables
Treasure Trove Hinterlands Prosperity Bank Monument Royal Seal Trade Route Venture Cache Develop Fool's Gold Ill-Gotten Gains Mandarin
Grim Parade Dark Ages Dark Ages Armory Band of Misfits Catacombs Cultist Forager Fortress Hunting Grounds Knights Market Square Procession
Watery Graves Dark Ages Seaside Count Graverobber Hermit Scavenger Urchin Native Village Pirate Ship Salvager Treasure Map Treasury
To the Victor Dark Ages Guilds/Cornucopia Bandit Camp Counterfeit Death Cart Marauder Pillage Sage Harvest Hunting Party Remake Tournament
Expert Intro Adventures Adventures Caravan Guard Coin of the Realm Haunted Woods Lost City Magpie Peasant Raze Swamp Hag Transmogrify Wine Merchant Mission Plan
Son of Size Distortion Adventures Base Bandit Bureaucrat Gardens Moneylender Witch Amulet Duplicate Giant Messenger Treasure Trove Bonfire Raid
The Hero's Return Adventures Guilds/Cornucopia Artificer Miser Page Ranger Relic Fairgrounds Farming Village Horse Traders Jester Menagerie Travelling Fair
Basic Intro Empires Empires Castles Chariot Race City Quarter Engineer Farmers' Market Forum Legionary Patrician/Emporium Sacrifice Villa Wedding Tower
Advanced Intro Empires Empires Archive Capital Catapult/Rocks Crown Enchantress Gladiator/Fortune Groundskeeper Royal Blacksmith Settlers/Bustling Village Temple Arena Triumphal Arch
Grand Scheme Intrigue Base Bridge Mill Mining Village Patrol Shanty Town Artisan Council Room Market Militia Workshop
Dusk Nocturne Nocturne Blessed Village Cobbler Den of Sin Faithful Hound Fool Monastery Night Watchman Shepherd Tormentor Tragic Hero
Midnight Nocturne Nocturne Conclave Crypt Cursed Village Devil's Workshop Druid Exorcist Leprechaun Pooka Raider Secret Cave
Illuminati Guilds/Cornucopia Alchemy Butcher Herald Masterpiece Merchant Guild Stonemason Apprentice Golem Philosopher's Stone Scrying Pool University
Tonics & Toxins Guilds/Cornucopia Alchemy Baker Candlestick Maker Doctor Plaza Soothsayer Alchemist Familiar Herbalist Transmute Vineyard
Overture Renaissance Renaissance Acting Troupe Experiment Flag Bearer Hideout Improve Inventor Lackeys Old Witch Seer Treasurer Fair
Prelude Renaissance Renaissance Border Guard Cargo Ship Ducat Mountain Village Priest Recruiter Sculptor Silk Merchant Swashbuckler Villain Citadel Star Chart
This also has an extra pair of card-shaped things, and is the only solution with 12, so far. Other solutions have an extra kingdom card, though (using Wine & Dine, which has a Herbalist bane).
Interestingly, all of the solutions so far include a pair of the new Guilds/Cornucopia + Alchemy sets; 6 of the 9 use the Guilds ones, while the other 3 use the Cornucopia ones. The Cornucopia pair includes every Alchemy card except for Possession.
Script is nowhere near done; I have the kingdoms ordered by a least-overlap metric, and the first thread is still checking combinations with kingdoms 1, 2, 7, 8 (Dark Ages, Nocturne, Nocturne, and Dark Ages single set kingdoms); 3-6 are all Nocturne kingdoms which overlap with 2. The second thread skipped set 1, and is still also on 2, 7, 8, though it is now on its second fourth set (20 - Big Actions - and now 23). The third thread started with 3, 4, and hasn't found a 20-kingdom solution yet.; at the rate it is going, it may finish checking the rest of the combinations before the first thread exhausts set 1; starting with two multi-set kingdoms seems to be messier in terms of eliminating other kingdoms more quickly, so fewer combinations to check.