"Life is Strange" was fun. The story got pretty dark towards the end.
So you play as this girl, Max, who develops the power to travel back in time to change things when she sees a girl get shot. She starts out using it for small things (outsmarting the girl who bullies her, showing off to her classmates...) but eventually Max uses her powers in more significant ways. All of your choices have effects (for example, if you are nice to the nerdy boy who does art you can convince him to step outside his comfort zone later in the game. if you are rude or ignore him all together, he might not listen to you. this is a minor example, i didn't wanna pick one that had major effects on gameplay.)
I got the game for $20 and id say its worth it, especially for the replay value. (making different choices and all).
MAJOR EDIT: forgot to mention this game is rated M for blood, intense violence, sexual themes, strong language and drug and alcohol use. like i said it gets pretty dark.
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The lego games are also always a good bet. We enjoyed lego jurassic world, but any that appeal to you is probably a solid choice.
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I would also recommend the newly released no mans sky. it is a first person, and its kind of a shooter? i don't like shooters either but i'm enjoying this one since the shooting is a very minor part. if you enjoy the fishing and collecting part of ACNL i'd say this might be up your alley. let me know if you wanna chat more about this (like this post or msg me or whatever)
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but anyway yeah we also just got a PS4 and I feel ya on the "too many shooters front" but there is a solid library of other games. these are the only ones i've played personally, but i'm very interested in "Journey", "Heavy Rain", "Two Souls" (these were all ported to the PS4), Gravity Rush, The Witness, Firewatch, Unravel (I played the demo for this and it was cute), tearaway, Abzu (made by the same people as journey).
Like i said if you wanna chat some more hit me up.

I have a busy couple of days at work so i may or may not get back to you right away.
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Holy long post Batman sorry about that