Alright, I have plenty of experience in finding and playin premium mobile games, so I'll help you out!
I'm just gonna get this out of the way, if you have Android, get Exit Fate, it's a freeware rpg maker game that is a love letter to the Suikoden series and is literally one of my favorite RPG's of all time, with an amazing story, great characters, a neat turn-based battle system and many party members to recruit all over the world, it's amazing. you can download it on Pc if you don't have Android, but it's on there for free too.
Anyway, my main experience is in finding RPG games, but I've looked for and enjoyed many different titles over my years of playing on the platform. There are a ton of mobile threads over the years to check out if you want to expand your search, and Touch Arcade also has many titles it has covered.
Since you are looking for games that require little attention, I'll go for what I know in non-action titles.
Snakebird is a charming puzzle game designed to be a vertical 2d puzzler version of the hit old game SNAKE, where you go through each of the levels guiding one or more "Snakebirds" by swiping in the direction you want them to go. The game is on a grid, and the snake birds are made up of a length of squares that follow gravity together, so if you aren't long enough to bridge a gap between two places, they'll fall, and if you angle yourself, you can move forward and around gaps and obstacles. It's a lot easier to play in practice, and the levels can get quite devious in design and mechanics later on. It's also Free to Try, Pay to Unlock the whole game, so give it a shot.
The Chaos Rings series are RPG's developed by the Wild ARMS studio Mediavision and published by Square Enix, and are traditional RPG games with a focus on partnered protagonists working together to fight monsters, either attacking as two separate characters, or combining both their strength to attack at the risk of both being damaged by single target moves. That's quite a simplification of the General battle system, but the main reason you'll be playing these games are the fantastic storylines. The first deal with a four group death tournament for immortality, where each pair is made up of a man and woman, each playable in their own special story. The second is about one man being surprisingly and tragically thrust into a position of sacrificing the important people his friend cared for, his friend being the first, for the sake of saving the world. And the third, a bigger departed in gameplay and general tone, is about a boy living in a space colony orbiting a dangerous yet beautiful planet named Marble Blue, wishing to take up his lost father's torch and becoming an adventurer.
They are all amazing games in their own right, and require not that much thought to enjoy their battle systems, while having depth and fantastic stories to play through. I couldn't reccomend them enough, especially the second and last one. None of them are successive stories, with the exception of a good prequel to the first game, Chaos Rings Omega, and all but the first have some level of enjoyable postgame content to enjoy. Get them if you can, you won't be disappointed.
It's been reccomended earlier, but the Dragon Quest Series, and nearly all the titles in it too, have been ported to the phone over the years, and they take to it fantasticlly. While I'm unsure exactly how well DQ8 plays on contrast to the others, all the DS titles and the classic Trilogy, meaning the first 6 games, have migrated amazingly, and have great phone functionality. They are all diamonds, and I would reccomend any DQ game you are interested, particularly the third, and the second trilogy starting with 4, with 5 being one of the best in the series. DQ 8 is also very good, but if you have the 3DS version or original, I'd likely play those first.
MewMew Tower, is a masterclass in "stack this stuff as tall as possible" games, not doing much to reinvent the genre, but being about adorable cats, with a birdy mode too, and having gyro intergration involving balancing the tower of cats. It's very simple and makes me laugh every time I play it, so get it for free sometime.
ReversiQuest 2, is a really strange but amazing game, essentially pegging itself as a cross between Tactics Ogre, and Othello, also known as Reversi, also known as Go. You explore maps in different chapters of the game, going down paths in two different directions and making your way through chests, random events and enemies as you reach the end, with different playable characters capable of traversing differently as leaders of the party. Once you start fighting an enemy, you become thrusted into a tactical RPG version of Go where you and your opponent each take turns placing units into a field of battle, converting lines of units to your side and dealing a variety of effects as the game goes on depending on which units do the flipping, such as two units with Green mana healing you for the flippers amount, or a red and a yellow unit flipping with another, attacking and gaining a resource to use abilities. It's complicated to say all at once, but the game really comes into it's own as you play it, and it's quite a fun game to enjoy once you try it. It's story is lightly interesting, and it's asthetic design is impeccably inspired by Tactics Ogre and FFT. It also has a TON of content, and gets more and more complex with it's units and playable characters as it goes on, with new chapters and improvements being made every once in a while. I'd say it's completely worth your money.
That's all I got now, but hopefully I can return with more games later, I hope you find a ton of good games to play OP, the phone has numerous Premium titles to enjoy that are buried and forgotten, and they need to be found!
