UPDATE: As DrunkenMessiah is no longer on the site I will try to regularly keep this as up to date as I can. If someone new gets added that I miss please feel free to post it in the comments or send me a PM. Thanks, Munch.
This is a list of everything free on the PSN store to all users feel free to add anything i've missed.
Check out this awesome guide on how to access other regions PSN Stores! How to: Access other regions PSN StoresImportant Game NotesSome titles appear to be free but you will need to purchase them to continue to play. So far the only free titles that you have to purchase to finish are:
Capcom Arcade Cabinet - You get one free game, the rest you have to buy.
Killzone 3 (Multiplayer Only) - Limits you to level 8-10, therefor unable to get Plat. It has different trophies to the original Disc version.
Uncharted 3 (Multiplayer Only) - Only lets you go up to level 15 in the free version.
PS4 Free GamesDC Universe Online (EU) (PS4) /
DC Universe Online (PS4) (can't obtain trophies in free version)
https://www.truetrophies.com/The-Pinball-Arcade-PS4/tro....htm (one free table, table trophies can be obtained)
https://www.truetrophies.com/Zen-Pinball-2-PS4/trophies.htm (one free table, trophies can be obtained)
https://www.truetrophies.com/War-Thunder/trophies.htm (has no trophies)
Blacklight: Retribution (has trophies)
https://www.truetrophies.com/Warframe/trophies.htm (has trophies)
US Only Free PSN GamesEverybody Dance: Digital (Trophy Supported) (Playstation Eye Camera required!)Jetpack Joyride (Trophy Supported)Jetpack Joyride (Vita) (Trophy Supported)Pinballistik (Trophy Supported)SingStar Digital (Trophy Supported)
Toro's Friend Network (Vita)
Zen Pinball 2
(Trial Free to Play Mode only!) (Has trophies if you upgrade to the full game!)Zen Pinball 2 (Vita) (Trophy Supported) (?) Conformation needed)US/Canada Only Free PSN GamesBack to the Future: Episode 1 (Trophy Supported)Japan Only Free PSN GamesJoysound Dive
(Unknown as to what this is as it is region locked!)Mobile Suit Gundam Battle Operation (Trophy Supported)Pawapuro Stadium
EU Only Free PSN GamesDeathmatch Village
Sam & Max Episode 1: The Penal Zone (Trophy Supported)The Walking Dead - Episode 1
PSN Games (PS3)Buzz! Quiz Player
Capcom Arcade Cabinet (Trophy Supported)DC Universe Online Free to Play
Defiance (Not sure if trophies unlock in this new free to play version)
DUST 514
Free Realms This has had its servers closed and is now unplayable
Go! Sudoku Starter Pack
Killzone 3 (Multiplayer Only) (Trophy Supported)PlayStation Home (Trophy Supported)Playstation Home Arcade
Spartacus Legends (Trophy Supported) (Not available for free in Germany.)Tekken Revolution (Trophy Supported)Uncharted 3 (Multiplayer Only)
UNO
(Trial Free to Play Mode only!) (Has trophies if you upgrade to the full game!)PSN Games (Vita)Crazy Market (Vita) (Trophy Supported)BreakQuest: Extra Evolution Freemium (Trophy Supported)Cliff Diving
Ecolibrium (Trophy Supported)Destiny of Spirits (Vita) (Trophy Supported)Fireworks
Frobisher Says (Trophy Supported)Imaginstruments
Pain Park
Paint Park Plus (Trophy Supported)Playstation Home Arcade (Vita)
Skype
(Not a game, but is listed on PSN as a game)t@g
Table Football
Travel Bug (Trophy Supported) (?) Conformation needed)Treasure Park
Treasures of Montezuma Blitz (Trophy Supported)Wake-up Club (Trophy Supported)Themes (PS3)
Athletics Dynamic
(PS+ Exclusive)Beach Dynamic
(PS+ Exclusive)Christmas Lights Dynamic
Chroma Dynamic
(PS+ Exclusive)City Drive Dynamic
(PS+ Exclusive)Destiny Dynamic
Dragon 2 Dynamic
(PS+ Exclusive)Fish Tank 2 Dynamic
(PS+ Exclusive)Gangster Squad Dynamic
Grid Dynamic
(PS+ Exclusive)Hangover 3 Dynamic
Hell Yeah! Dynamic
(PS+ Exclusive)House of the Dead 4 Dynamic
(PS+ Exclusive)Jak and Daxter Sketchbook Dynamic
(PS+ Exclusive)Jet Set Radio Dynamic
(PS+ Exclusive)Mad Riders Dynamic
(PS+ Exclusive)Man of Steel Dynamic
Mesh Dynamic
(PS+ Exclusive)ModNation Dynamic
MotorStorm: Arctic Edge Dynamic
MUBI Dynamic
Ni no Kuni - Wrath of the White Witch
(PS+ Exclusive)Oddworld Dynamic
(PS+ Exclusive)Pacific Rim Dynamic
PlayStation Plus 1st Anniversary Dynamic
(PS+ Exclusive)PlayStation Plus 2nd Anniversary Dynamic
(PS+ Exclusive)PolyWave Blue Dynamic
(PS+ Exclusive)PolyWave Red Dynamic
(PS+ Exclusive)Samurai Dynamic
(PS+ Exclusive)Skaters Dynamic
(PS+ Exclusive)Start the Party! Dynamic
Swirlpool Dynamic
(PS+ Exclusive)Tales of Graces f Special Custom
The Traveller Dynamic
(PS+ Exclusive)Vodafone
Winter Journey Dynamic
(PS+ Exclusive)Wipeout HD Dynamic
World Cup Dynamic
Misc (Vita)
Ridge Racer Music Pack 1
Ridge Racer Music Pack 2
Ridge Racer Music Pack 3
Ridge Racer Music Pack 4
Ridge Racer Music Pack 5
Ridge Racer Music Pack 6
Ridge Racer Music Pack 7
Ridge Racer Music Pack 8
Ridge Racer Music Pack 9
Ridge Racer Music Pack 10
Ridge Racer Music Pack 11
Ridge Racer Music Pack 12
Ridge Racer Music Pack 13
Ridge Racer Music Pack 14
Ridge Racer Music Pack 15
Ridge Racer Music Pack 16
Ridge Racer Music Pack 17
Ridge Racer Music Pack 18
Ridge Racer Music Pack 19
Ridge Racer Music Pack 20
Thanks to Gaelic-_-Flame, Shadow_Enz, munchmagic1986, alklein92201, DARTHWOLFEN, finallife5, Linkx41 & RyuDc for helping out with the list!