Ingress
It's a game where you have to physically move around in to get to portals, capture or further fortify them for your own team (two factions, the green Enlightened and the blue Resistance, the only real difference between them besides the colour is how many people play them locally and what kind of community/culture they have built). You can attack and defend portals, capture them, link them together, make fields for MU (Mindcontrol units, also mackerel units if its over water), there are check points every 5 hours for scoring fields but you don't have to care about that if you don't want to. It's fairly easy to reach level 8 which is when you have access to all of the gear, the levels beyond that are just a bonus.
You get to explore your city, discover lots of art, statues, architecture, history, parks, unique local businesses, random stuff that the portal monkey google interns approved even though it shouldn't have been, etc. Also there are missions you can do that often further encourage exploration in your city, they also have mission badges and some people have created dozens of missions that when done in the correct order result in all of the badges forming a picture on your profile.
You get to walk around, get fresh air and exercise (I know several people who have lost weight playing this game, and one guy who has some major heart related health issues that occasionally force him to use a wheel chair, who has seen a huge improvement to his physical and mental health and managed to hike up mountains he wouldn't have thought he could have two years ago).
You can also play as much or as little as you want to and meet up with as many or as few local players as you want to (we have a number of local players who prefer to play by themselves, and one who seems to be terrified of meeting anyone). Many cities hold a monthly Ingress First Saturday event where they have a cross faction meet up, its mostly sanctioned win trading and helping new players level up. Our local one is pretty fun, even have various swag prizes. Both teams here also use the Slack chat app for organization, planning fielding ops, and general chit chat and cat picture sharing. We also occasionally have cross faction 'beergress' nights and seem to be the only people who attempt cross faction farms (generally not very successful though).
Otherwise... no idea. This is pretty much the only game I play on my phone, although Fallout Shelter is worth checking out and fun for a week or so.
I'm not the best at explaining things but its free and you should check it out. They did just add a store but you don't need any of the three items currently available, one of which is purely cosmetic.
