For the most part phone games suck. They are either repetitive, pay to win or so full of ads that they are unplayable.
So what are your recommendations? I don’t mind buying a game if it’s worth it but I don’t want a pay to win game.
I was having a lot of fun playing Hitman Sniper on my iPad for awhile. I think I only paid $5 for it. I don’t know how it would be on a phone, but the screenshots in the link show it as if were played on a phone:
XCOM 2 is a very well implemented port.
There’s also an interesting branch of NetHack called GnollHack.
See a prior thread:
Although you didn’t specify Android or iPhone, or your tastes in game.
But if you want free… yeah, lots of poor options, although I still think Fallout Shelter is fun with minimal PTW.
eta - simulpost with Asterion, as I also mentioned Xcom in the old thread. 
People love Shattered Pixel Dungeon.
What means “mobile game”?
Pretty much any game can download thru a App to a smart phone.
Forge of Empires is good if you have loads of time over a long period. It’s never ending. Lots of challenges and moving parts.
Beware, it can take over your whole day.
It can go on a phone, tablet or laptop. I play on my phone and tablet. I see it better on the bigger screen so I usually do that.
ETA…it’s free. You can spend money. But don’t. You can be just as successful not buying their extras. They are very careful not to drive you crazy trying to sell you stuff.
Monument Valley
Simon Tatham’s Puzzles
The Room and its sequels
It’s old but the original Simple Rockets has been my go-to time killer since it came out. They have an updated version but that is more ‘complicate rockets’ than Simple Rockets 2 (now called Juno New Origins).
I also killed hours on a plane with Polytopia+. sort of a mini-civilization game which works well on a phone.
I am very much enjoying Forward Escape the Fold at the moment. An older game I also like is You Must Build a Boat.
I like Marvel Puzzle Quest (a free-to-play match-3 game where you activate powers based on your team of Marvel heroes/villains). I don’t know what you consider repetitive, but there’s a big variety of characters to unlock which provides me with a decent amount of variety.
Oh, figured I’d mention an old classic game originally for full sized systems that’s now been available on mobile for some time (although lots of different responses on how well it works from great to utterly broken).
Which is the full game we once played on PC (in my case) showing how much more powerful mobile devices are these days. I can say (leaving port questions aside) that it was a fantastic game in it’s day but can’t speak to all the people who feel that the port is excessively buggy.
And I don’t think I’d advise playing on most pocket-fitting phones, so unless you wanted to play on a tablet or phablet, may not meet the OP’s needs.
I do keep thinking about getting it, because of my fond memories of playing on the PC, but just never do so because I’m just not into most mobile gaming.
Well, I play Netflix’s Solitaire on my iPad, but could play it on my iPhone if I so desired.
Games on my phone I actually play:
Golf On Mars is a simple 2D golf game. Set on Mars, so gravity is weak.
2048 is a number grid adding/matching game. You start with two 2s on a 4x4 grid; sliding two 2s together produces a 4, sliding two 4s together produces an 8, and so on. Goal is to have a tile reach 2048. I haven’t done it yet. I’ve gotten one 1024 on screen once, that’s the closest I’ve ever gotten. (tip: pick a direction and never, ever slide in that direction unless you have no other choice)
Bewilderword is a word game. Blank boxes representing words arranged in a crossword style. You have 9 letters with which to form words of any length from 4 to 9 letters, using each letter only once, and one of those letters must be included in every word.
Pretty sure they were all free, all good for brief sessions.
The most impressive mobile game I know is Terraria, which has been developed/updated for years, alongside the PC version (with consoles as a bit of an afterthought). It astounds me every time I play it on my phone (even though some of the world-building is a bit whimsical for my tastes).
I have yet to find a game that wasn’t made specifically with playing on a phone in mind to be any good. A port of a game that was meant to be played on a bigger screen with a controller or a mouse tends to suck. I’ve downloaded and deleted a bunch of those.
I’ve played these for years after they were recommended on the Dope.
I’m replaying this, there’s some added content from the show.
If you have an Android device, fo2.exe allows you to play the first two PC games on your phone. You need some of the resources from a legally owned copy of these, but you can pick it up cheap on Steam or GoG.
Interesting, I own both F1-2 on steam… as I said in the prior thread, I wouldn’t want to play those games on the screen of my phone, but think it would be fine on my 10’ tablet.
If you have an iPhone, you can subscribe to Apple Arcade. There is a monthly fee, but it gives you access to a bunch of high quality games especially remade for Apple Arcade. There are no added costs, included no pay-to-win “features”.
The Apple Arcade cover all genres. The best part is you can play any and all of the games. Find the ones you like, uninstall the ones you don’t. Most people can find something they like.
My favorite is Bloons, which is a tower defense game with monkeys popping balloons. The version available in the general App store has pay to win, but the Apple Arcade version does not. Lots of fun.
I played FoE a LOT for a number of years. Yeah, it can take over.
One of the good things about it though is that you can play it on your phone, your tablet, or your computer screen, and it’s all the same account. Some other games (e.g. Bloons Tower Defense) would be completely different games, your progress would not transfer. I did occasionally spend money, but mostly used Google Play credits I got for free - and you never HAD to spend the money.
Lately I’ve been playing a variety of ad-sponsored games where if I like them enough, I can spend a few dollars to make the ads go away. The ones I hate are ones where the payment is a subscription - they are likely hoping you’ll forget it and keep paying. I won’t sign up for those.