As of September 21st, 2018:
9,908 Different Cards to Own (the top possible, including 12 that are unobtainable).
If anyone is curious about their Collection Statistics, the stat that says "Different Cards Owned" would top out at 9,908 (at least until the next set or more promos come out). This includes all sets through Dragon Majesty, and all types of cards (regular, reverse foil, promos, secret rare, trainers, energy, etc.)
There are a dozen "unobtainable" Promos, so nobody will have them all (see list below).
But for example, if you have 8,000 Different Cards Owned...
9,908 minus 8,000 equals 1,908 to go.
8,000 divided by 9,908 equals 80% complete.
NOTE: I'm fairly confident this number is correct, comparing my collection with several others that are nearly complete. If you've got a slightly different count (very close, but not exact), there's a glitch in the servers that we've discovered. The game statistic "Different Cards Owned" will keep track of cards that you previously owned, but may have traded away (it doesn't subtract them). To get an accurate number you need to replace those cards (it won't count them a second time). See post #34 for details.
Set Completions: We also tested this feature. Very similar to the "DCO Stat," once you've completed a set (in Collection Stats), that set's symbol remains Complete even if you trade some of the cards away. See post #77 for details.
These are the 12 Promo cards currently unobtainable (and not tradable) in the game.
EDIT: Changed the overall count to 8,801 (rather than 8,802). See post #4.
EDIT: With another League Promo, the count is 8,802 again. See post #10.
EDIT: With six new cards added to the game, it's now 8,808. See post #16.
EDIT: A new Marshadow Promo added to the game. Total 8,809. See post #25.
EDIT: The version 2.50 Update (on January 30th) added 7 Theme Deck exclusive cards from the two Call of Legends decks. This would probably be considered a bug fix, rather than new cards being added. Total 8,816.
EDIT: The new set, Ultra Prism, added 303 cards, and 9 more promos were added (312 altogether). This brings the overall total to 9,128. See post #33.
EDIT: A promo Darkrai-GX added to the game. Release 3/9/2018. Total 9,129. Post #36.
EDIT: A League Promo, Infernape UPR 23, was added. Total 9,130. See post #45.
EDIT: A promo Lucario-GX added to the game. Release 3/9/2018. Total 9,131. Post #46.
EDIT: Two new Necrozma-GX promos (Dusk Mane & Dawn Wing) from the Spring 2018 Tins were added to the game with the v2.51 update. Total 9,133. Post #49.
EDIT: Three promos (Dusk Lycanroc, Dusk Mane Necrozma, and Dawn Wings Necrozma) from the Spring 2018 Collector Chest were added to the game. Total 9,136. Post #50.
EDIT: A League Promo, Empoleon UPR 34, was added. Total 9,137. Post #51.
EDIT: A new Legends of Johto promo box with Entei GX (Alt A, SLG 10a) and Raikou GX (SM121) was added. Release 4/6/2018. Total 9,139. Post #52.
EDIT: Four promos from S&M Trainer Kit. Alolan Sandslash (SM127) and Alolan Ninetales (SM128) come in both Foil and Non-foil versions. Release 4/20/2018. Total 9,143. Post #53.
EDIT: Forbidden Light and the v2.52 update added 266 new cards (254 for FLI, and 12 others). This brings the overall total to 9,409. See post #56.
EDIT: A Zygarde-GX promo (SM122), released 6/1/2018, and a Guzzlord League Promo (FLI 80) were added. Total 9,411. Post #59.
EDIT: A League Promo, Beast Ring FLI 102, was added. Total 9,412. Post #60.
EDIT: A new Forces of Nature GX promo box, with Thundurus GX (SM133) and Tornadus GX (SM134) was added. Release 7/6/2018. Total 9,414. Post #61.
EDIT: The version 2.54 update (7/31/2018) added a foil Promo Pikachu (BW54), source unknown, and deleted two cards from the database. Total 9,413. See post #67.
EDIT: Celestial Storm and new promos added 335 new cards (324 for CES, and 11 others). This brings the overall total to 9,748. See post #68.
EDIT: The glitches that appeared in the v2.54 update were fixed. A Pikachu Promo was removed, and two missing cards were restored. Total 9,749. See post #69.
EDIT: The 2018 World Championships gameplay set was added, which brings 6 more Champions Festival Stadiums in various languages. Total 9,755. Post #70.
EDIT: The v2.55 update added a League Promo, Metagross CES 95. Total 9,756. Post #71.
EDIT: The set Dragon Majesty added 134 cards (116 Pokémon & 18 Trainers). This brings the overall total to 9,890. See post #73.
EDIT: 14 cards were added (9/7/2018). Two are DRM promos (Latias & Latios) and 12 are alternates from the upcoming Battle Arena Decks. Total 9,904. See post #74.
EDIT: Four promos were added (9/21/2018). Naganadel-GX is in DRM Elite Trainer Box, and 3 new Eevee Tins include Glaceon-GX, Leafeon-GX, and Sylveon-GX. Total 9,908. Post #75.
Edited by RobRatt, Yesterday, 08:24 AM.