This app was mentioned in 43 comments, with an average of 1.44 upvotes
Maybe you'd like Outwitters. It's a turn based strategy game, but you can play it in an asynchronous online multiplayer mode. You can have multiple games going at once. It works with you doing your turn and then you can go do something else. Your opponent receives what you did, then can think about it and respond. You check back later and can see what they did and then take your turn. If they basically quit by not playing anymore, you'll end up winning.
There is 1v1, 2v2, leagues, and there is also a local multiplayer mode where you can pass the phone back and forth between a friend. It's available on iOS too, so it's good if you want to play with a friend who has an iPhone.
It's important to note that this game is not pay to win / p2w. The game is free with ads for you to check out if you like it. If you get any of the one time purchase IAPs, it removes the ads. There are one time purchase IAPs for things like raising the maximum number of games you can play at one time, slightly different groups to play, etc. The slightly different groups are all almost the same and anyone can get them, so you don't have to grind forever or spend tons of money on stuff like opening random chests for maybe a chance at an upgrade.
Outwitters is great. You can play and take turns whenever you can, solid fun strategy, one time purchase to get everything, with a free version that just lets you use less of the factions. Haven't played in years, but it sounds like it could fit for what you're looking for.
Try looking into games with online, asynchronous, multiplayer.
It might not meet all the criteria you mentioned, but Outwitters let's you have multiple games going with multiple opponents. You can take your turns whenever your connection is working.
Outwitters might fit the bill. It's a turn based strategy game that's solely correspondence based. It's free to play 5 games at a time, but if you buy any of the IAP's, that unlocks up to 25 games at a time, which is plenty. I really enjoy it so you might want to check it out.
If you're looking for something really hardcore, you could check out /r/baduk and then proceed to lose your life to that game, just like me!
You good with tile based SRPGs?
LinkMe: Uniwar, Outwitters.
They both have online multiplayer.
Uniwar looks like an old GBA game with factions reminiscent of Star Craft.
Outwitters is a cutesy strategy game with 4 factions, but 3 of them are IAPs. The only difference between factions is their special unit that is tied to their theme (Plant faction has a Flower that can block tiles with thorns, Aquatic faction has a snail that acts as a cannon and is the only source of AoE damage in the game, Robot faction has a mind controller, Cutesy faction has a whale that can teleport units next to it)
EDIT: Bot got the wrong game. here it is: https://play.google.com/store/apps/details?id=com.onemanleft.outwitters
Sorry I don't have any reviews, I have no experience with any of them.
Outwitters. Don't let the art style put you off. It plays like Starcraft + Chess.
Aside from the art direction, I actually prefer Outwitters over Uniwar. This is mostly because the way units attack and do damage is much simpler. You can quickly see how much damage your units will do instead of having to calculate it every time which means you spend more time focusing on the strategy and less on doing basic math.
Official Website | Trailer | Replay Website | Android App Store | iOS App Store
I would look at the replays if you want to get an idea of how the game plays. Note that this is not what the game actually looks like graphically. Watch the trailer or look at the app store screenshots to see what the game graphics look like.
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One game I played quite a bit was Outwitter. Fun game for me, but not sure if it's exactly what your looking for. https://play.google.com/store/apps/details?id=com.onemanleft.outwitters