I have a lot of games that I can tell you to steer clear of
Unfortunately most mobile games are shameless pay2win in app purchase pushers or just lack proper controls/gameplay elements to be worthy of extended time. It also kind of depends on what you are trying to find. Do you just want something you can flip on while waiting in your dentist's lobby for 15 mins or are you looking for an engrossing experience and trying to use your phone in lieu of a nintendo 3ds or a laptop? Cus for me it's only the former, so that kind of taints me as well since I am not looking for a really great 50 hour game on my phone.
But here's some I have played on iphone.
Great little war game - this one is decent enough. It's kind of like advance wars but simpler. I got great big war game on steam really cheap and liked it a lot. Sadly the ios one has far less features like no aoe damage which really dumbs down the gameplay. It's ok.
Galaxy on fire 2 - This one was a good space sim that suffered from terrible controls. I died on the easiest settings on like the 3rd mission cus the controls sucked so badly. If you have like a controller or something for your phone this one might be great. Reminds me vaguely of old x-wing games.
Braveland - This one is very cool until you realized it's a super dumbed down version of homm. Literally one path to take, completely linear, no leveling, no upgrades. Could've been a great game with just a tiny bit more openness added.
CivRev - Really simple civ game. It's ok for a time waster. No serious strategy elements of course.
Angry Birds - Everyone's played this by now but this game is awesome. It's every thing I want out of a mobile game, great controls, easy to use, but it's actually fun to the point I might plop down on my couch and play it for an hour to relax vs firing up the PC. It's actually fun!
Game dev story - This one was actually a lot of fun for me and worth buying. When I first picked it up it was a blast trying to unlock new game combinations and level up my staff and get a hit sale. Of course it's not realistic in the least and not much of a sim. I mean it's like impossible to lose money and you don't really do anything, just pick a game title and wait for it to be done. But I found it strangely addicting. It doesn't have replay value though cus since you don't do anything there's really no point in experiencing it again, no different paths to take really, just try to make more money I guess.
Candy crush - I have spent zero money on this game, but it's actually one of my favorite ones. If you like brick breaker type games at all give it a shot. It's really fun to play for a quick 5 min break between meetings and stuff. The randomness of it does make you question how much skill is really involved but idk, it occupies me during downtimes just fine.
Other than that I love sudoku and spider solitaire as well.