Fat Princess Adventures, Diablo 3, that Tetrisy tower game from last month, Amplitude, Don't Starve Together, Broforce, Helldivers, Super Time Force...

Magicka 2
Fat Princess Adventures
Lara Croft temple of osiris
LBP 3

Diablo

Alienation

Borderlands

Guacamelee (2player)

all the others are 4player local.

just some from the top of my head.

Altough the Xbox really wins here (Master Chief Collection, Gears of war, Van Hellsing)

There are some fun games on the PS4 too, here are a few I played with my friends and enjoyed:
The Gauntlet
Diablo 3
Octodad
Overcooked
Dead nation
Alienation

Especially the housemarque games are a lot of fun. And also Beyond: Two Souls has a remastered version andyou can actually co-up this, one being the girl and the other the ghost, I didn't like the game much but it was a lot more fun in co up :)

Overall if local multiplayer is your cup of tea in all honesty I do recommend both Microsoft and Nintendo consoles over the PS4 since a lot of their AAA-games are co up

Enter the gungeon
Alienation
Gauntlet Legends
Black Ops 3 (for zombies, some good DLC content too)
Rock band 4


Top 3 are indies. I highly recommend enter the gungeon anove the others though.

Towerfall
Towerfall
Towerfall
Towerfall

...
and The Dark World Expansion adds 4p coop

...
Seriously, get Towefall and extra controllers!

Also, some others:

Trine 1,2,3 (3p coop)
Gauntlet
Lara Croft Temple of Osiris
Magicka 2
Never Alone
Guacamelee (4p coop platformer!)

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IF you really wish great games of this tipe, a WiiU would be a great aquisition, with the Mario Games (4p local), Smash (8 player local!!!) and Nintendo Land (5 player local) you will get a ton of fun for sure.