I'll still update if Sony will not drastically speed up its current "few games per quarter" tempo since the thread is a convenient place to discuss the related issues, but do not hold me to it. :=)
THE LIST (current as of 11.09.2017)
ADK DAMASHII (US, UK, DE)
Arc the Lad: Twilight of the Spirits (US, UK, DE)
Art of Fighting Anthology (US, UK, DE)
Ape Escape 2 (US, UK, DE)
Bully / Canis Canem Edit (US, UK, DE)
Dark Cloud (US, UK, DE)
Dark Cloud 2 / Dark Chronicle (US, UK, DE)
Destroy All Humans! (US, UK, DE)
Destroy All Humans! 2 (US, UK, DE)
Eternal Ring (US, UK, DE)
Fantavision (US, UK, DE)
Fatal Fury: Battle Archives Volume 2 (US, UK, DE)
Forbidden Siren / Siren (US, UK, DE)
FU'UN SUPER COMBO (US, UK, DE)
GTA III (US, UK, DE)
GTA San Andreas (US, UK, DE)
GTA: Vice City (US, UK, DE)
GTA Trilogy (US, UK, DE)
Harvest Moon®: A Wonderful Life Special Edition (US, UK, DE)
Harvest Moon®: Save the Homeland (US, UK, DE)
Hot Shots Tennis / Everybody's Tennis (US, UK, DE)
Indigo Prophecy / Fahrenheit (US, UK, DE)
Jak and Daxter: The Precursor Legacy (US, UK, DE)
Kinetica (US, UK, DE)
Manhunt (US, UK, DE)
Max Payne(US, UK, DE)
METAL SLUG ANTHOLOGY (US, UK, DE)
OKAGE: Shadow King (US, UK, DE)
PaRappa The Rapper 2 (US, UK, DE)
Primal (US, UK, DE)
Psychonauts (US, UK, DE)
Puzzle Quest: Challenge of the Warlords (US, UK, DE)
Red Dead Revolver (US, UK, DE)
Red Faction (US, UK, DE)
Red Faction 2 (US, UK, DE)
Resident Evil™ Code: Veronica X (US, UK, DE)
Rise of the Kasai (US, UK, DE)
Rogue Galaxy (US, UK, DE)
Samurai Shodown VI (US, UK, DE)
Star Ocean® Till The End Of Time™ (US, UK, DE)
Star Wars Bounty Hunter (US, UK, DE)
Star Wars Jedi Starfighter (US, UK, DE)
Star Wars Racer Revenge (US, UK, DE)
The King of Fighters 2000 (US, UK, DE)
The Mark of Kri (US, UK, DE)
The Warriors (US, UK, DE)
Twisted Metal Black (US, UK, DE)
War of the Monsters (US, UK, DE)
Wild Arms 3 (US, UK, DE)
Upcoming :
Ape Escape 3
Legend of Legaia 2
Harvest Moon: A Wonderful Life Special Edition
PS2 Remasters
Since few high profile PS2 games are also available on PS4 in form of remaster, and not just running on the emulator, we decided to add them in a separate list:
Final Fantasy X/X-2 HD Remaster (US, UK, DE)
Final Fantasy XII: The Zodiac Age (US, UK, DE)
Cel Damage HD (US, UK, DE)
Kingdom Hearts HD 1.5 +2.5 Remix (US, UK, DE)
Legend of Kay Anniversary (US, UK, DE)
Yakuza Kiwami (US, UK, DE)
Upcoming remasters:
.hack//G.U. Last Recode
Shadow of the Colossus
Yakuza Kiwami 2
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Want your favourite PS2 game on PS4? Get active, Sony is looking for suggestions.
Please say if I missed anything, I'll try to correct asap! Was considering adding the prices, but there are deals/PS Plus discounts now everywhere, so it would look complicated. There are basically two price tiers (9.99€/$ (7.99£) and 14.99€/$(11.99£)) with bigger games being more expensive, of course.
Many thanks to Jackstin for compiling the remasters list.