As a potential upper bound:
*have exactly 17 known, distinct cards in your draw pile, including a Stash, and all from the Black Market deck (or Shelters).
*play University, gain the Inn. Shuffle, placing the Stash 7th from the top.
*play Navigator, looking at the top 5 cards and returning them.
*play 10 Pearl Divers, bringing all 10 cards to the top. This leaves Stash at the bottom, and the card just above it we know by process of elimination.
*gain and topdeck (with Watchtower) 43 unique cards (10 knights, 9 other kingdom cards, 1 bane, 4 base victories, 5 base treasures, 1 Curse, 5 Ruins, 5 Prizes, 1 Spoils, 1 Mercenary, 1 Madman)
*play 10 Black Markets and 10 BoM as Black Market to gain and topdeck 20 unique cards
*topdeck 10 cards with Count
*topdeck 10 cards with Mandarin
*topdeck 10 cards with Courtyard (topdeck 3 nonuniques by gaining and topdecking with Watchtower, then play Courtyard to draw them and put back a unique. Repeat)
Total: 17+43+10+20+30 = 120 differently named cards.
Edit: 10 more from Scavenger. However, since I'm using these full stacks of 10, I cannot include as topdecked: Pearl Diver, Black Market, Band of Misfits, Courtyard, Mandarin, Count, Scavenger. So that's -7 cards. Thus my revised upper bound is 123.
Edit: 10-1 more from Graverobber. Total: 132
Kingdom:
Black Market, Pearl Diver(bane), Band of Misfits, Courtyard, Mandarin
Count, Scavenger, Inn, Tournament, Graverobber, Knights
Edit: Oh, 16 more by playing one of each kingdom treasure (purchased ahead of time from the Black Market deck. doesn't include Diadem, Spoils, or Stash) before gaining a Mandarin from the Supply.
Total: 148.