+10 points to The Last of Us, Overcooked and Lovers in a Dangerous Space Time (which I can now no longer pronounce correctly thanks to Matt and Quinn's Let's Play).
Helldivers and Guacamelee are my other local co-op recommendations, a top-down team shooter and a metroidvania-esque beat-em up respectively.
In terms of games that are PS specific, I'd definitely recommend Bloodborne if you're into the Dark Souls genre. For more light-hearted adventures, I enjoyed Tearaway Unfolded for its interesting use of controller mechanics and papercraft art style, and it's got the best introductory cutscene I've ever seen. (It involves TV guides and telenovellas!) Journey is a beautifully sweeping landscape and an examination of how deeply you can connect with a stranger over a couple hours of not-talking. If you like puzzle platformers and Lost Vikings style character switching, then -klaus- is another solid indie game with an interesting art style and strong mechanics.
Also it's not a PS exclusive, but I just recommend everyone play Firewatch.