I am listing USA titles only here. I do not use the Europe or other region stores.
USA store Full free games as of January 17th
Frobisher says! (Vita)
Fireworks (Vita - Augmented Reality)
Cliff Diving (Vita - Augmented Reality)
Table Soccer (Vita - Augmented Reality)
Treasures of Montezuma Blitz (technically free since you get more lives and crystals each day)
Lemmings (Playstation mobile - this is a 30 level Introduction to the game that plays much like the full game, but offers to sell you more levels.)
Bullion Blitz (Playstation mobile)
Jetpack Joyride (Vita)
Wake Up Club (this is a vita app, not really a game, but you can compete and earn trophies)
Treasure Park (Vita app)
Games that are free for a limited time only, being released on Wednesdays. (USA)
Samurai Beatdown (PSM - only free for 1 week) **This game may no longer be available.
Twenty Minutes Into the Future