In no particular order:
1. Avabel Online (Free)
Gorgeous 3D MMORPG for iOS and Android, free to play with a very active community and frequent updates. The game's structure is similar to Monster Hunter although most of it's still in Japanese and there are quite a lot of translation errors in the English version.
+ Beautiful graphics
+ Wide variety of activities, characters, and locations
+ In-app purchases are entirely optional
+ Supports external controllers
- Limited social features
- Translation errors
- Your account can only be linked to one device at any given time
2. Chrono Trigger/Final Fantasy VI/VII/IX ($4.99/$7.99/$10.99/TBA)
No introduction needed here, a fine selection of Square Enix JRPG titles ported to iOS and Android. FFIX has been announced and is slated for a 2016 release.
+ Solid gameplay
+ Additional features such as cloud-based saving
+ Enhanced graphics
+ Supports external controllers
- Expensive, even for ported titles
- Touch controls can be fiddly
3. Taiko no Tatsujin (Free + IAP)
A music rhythm game available only in the Japanese App Store, it features cute characters dancing while you drum away to popular songs from games, anime, and pop. Has a massive song library, but only comes with 5 tracks when initially downloaded. Overall it can cost a lot when the in-app purchases add up.
+ Fun to the point of being addictive
- Song pack costs add up
4. Assassin's Creed Identity (Free)
Plays very much like Assassin's Creed II, but in an episodic format. It also features wireless controller support, which is nice. Currently only available in the Australian/New Zealand App Stores.
+ Stays faithful to the franchise
+ Detailed, immersive gameplay
+ In-app purchases are entirely optional
+ Supports external controllers
- Crashes often
- Needs to be constantly connected to the Internet
5. Oceanhorn: Monster of Uncharted Seas ($4.99)
An original adventure RPG inspired by the style of the Legend of Zelda: Wind Waker. The main quest involves exploring numerous islands across a vast ocean and gathering artefacts from dungeons.
+ Enjoyable gameplay
+ Supports cloud-based saves across iPhone, iPad, and ATV
+ Beautiful graphics
- Limited side-quests to explore