As of April 6th, 2018:
9,139 Different Cards to Own (the top possible, including those no longer obtainable).
If anyone is curious about their Collection Statistics, the stat that says "Different Cards Owned" would top out at 9,139 (at least until the next set or more promos come out). This includes all sets through Ultra Prism, and all types of cards (regular, reverse foil, promos, secret rare, 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...
9,139 minus 8,000 equals 1,139 to go.
8,000 divided by 9,139 equals 87.6% 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 bug 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.
These are the 12 cards currently unobtainable (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 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.
Edited by RobRatt, 15 April 2018 - 09:58 PM.