I'm gonna recommend the big three that I recommend anytime someone is looking for amazing Mobile games, and then I'll probably toss out a few other titles I'd recommend.
1. Squids and Squids Wild West. These are physics flinging SRPG games (imagine final fantasy tactics only you just kinda fly across the map rather than using a grid) that star a bunch of different Squids, falling under 1 of 4 jobs (Scouts are great for exploration, Shooters have ranged attacks, Healers can heal, and Troopers can send out shockwaves to hit groups of enemies) all the characters are well defined, the graphics are gorgeous, the story is VERY good, which you may not expect from a series like this, and it uses an interesting Hat based level up system that lets you transfer stats to your character via hats, but it's a permanent upgrade so you can wear whatever hat you want after you get the stat boost. This is a great series, and while you can get it on PC, 3DS, and Wii U now, I'll always remember it fondly as an amazing mobile game where it started. There's a lot to do in this game, so I recommend it.
2. Nihilumbra. If you like 2D platformers that are wonderfully atmospheric, have a decent story (a few typos here and there though) and have a cool color mechanic that works out really well, look into this game. I had a lot of fun with it. I can't boost this one too much without giving stuff away, but I'd at least look into it.
3. Pretty much anything by [adult swim] They make a lot of great mobile games, so I'm just gonna toss out titles now. If any of these sound fun just based on their name, give 'em a look.
-Monsters Ate My Condo (and Super Monsters Ate My Condo)
-Bring Me Sandwiches!
-Castle Doombad
-Extinction Squad
-Snoticles
-Super Mole Escape
-Giant Boulder of Death
-Day of the Viking
-Delivery Outlaw
-Adventure Xpress
-Heads up Hot Dog
-Girls Love Robots
and I'm probably missing a few, but that's a fair list of stuff they have on offer.
And now I'm just going to name a few games I personally enjoy and I'll try to give a little bit about them to see if any of them appeal to you
Shin Megami Tensei - Not really a mobile game, but the first official release of the original SMT is exclusive to iOS devices. If you like old school challenging JRPGs, maybe give this one a look.
Detective Grimoire - A fairly fun (though admittedly somewhat simple) puzzle game where you play a Detective investigating a murder out in a swamp tourist attraction. The gameplay is pretty simple, but it's the animation and voice acting that makes this game stand out. It's more of an interactive story than a full game, but a good one. This was made by the Super Flash Brothers if you were ever around the old Newgrounds community
Endless Boss Fight - It's just what it sounds like. You're a little robot with boxing glove hands punching a bigger robot that will get stronger and stronger until it eventually kills you. The appeal on this one may wear out quickly, but I'm pretty sure it's free, so I'd at least give it a look. It's surprisingly addictive.
Fiz - A retro graphics game where you make beer. And that's it. Run your own brewery, sell to distributors, enter beer offs. I don't play a lot of these company simulations so I dunno if this one is much better than some of the others, but I had fun with it, and you can learn quite a bit about beer too if you're interested.
Hungry Oni and Jetpack Joyride - I'm not sure why I'm mentioning these two together, they're very different games, but both are cute, simple, and addictive. Jetpack Joyride came out in 2011 and I still go back to it every now and then. Both are also Free so I'd certainly give them a look.
Knights of Pen and Paper - Dungeons and Dragons where you don't only control the characters, but the players playing the characters. Fun game, I recommend it. Not much else to say on this one.
Layton Brothers: Mystery Room - Do you like murder mysteries? Do you enjoy stuff like the Ace Attorney series? Than this is a fun title for you. Unfortunately I have not beaten this game yet. My iPod Touch is rather out of date and tends to crash in the same spot near the ending. This one is free to download, you get to play 2 full cases, and then you pay like 5 bucks to unlock the rest of the game. So if it appeals to you at all, you're welcome to try it out.
Middle Manager of Justice - Another simulation game, this time with combat elements. You run an office that deals specifically in superheroes. Train your heroes and keep the city crime free. Now this one has some freemium issues later in the game, but it never really feels like a cash grab like some games do. Also if this matters to you at all, this is from Double Fine, so...y'know...
Honestly, I could keep going and going, but you probably want me to shut up. I'll leave you with these games for now, but I know a lot of other great mobile games if you're interested. Feel free to get in touch with me, lemme know what kind of games you like, and I'll go through my app library to try and find something for your tastes.
I'm weary of action games etc, because IOS controls aren't the best, but if there are some that defy that, then by all means 
