DungeonDiary is a lot of fun!
https://play.google.com/store/apps/details?id=jp.windbellrrr.app.dungeondiary&hl=en
I downloaded it randomly and I had no idea what I was doing at first. It's a dungeon crawler, but it's a little different. How it works is that you equip your little character and set her AI up with certain conditional logic rules. Then you pick a quest and set her off on her own in the dungeon, which she plays through automatically, following the rules you set up. Then she comes back, you identify your loot and upgrade your items, and you go back at it again. The dungeon crawling can be backgrounded, which lets you interact minimally with the game and still crawl through long Shiren-style dungeons. That bit is genius. And you get a widget to put on your homescreen with your character's current equipment.
You can customize your equipment with stat-ups and different colors. There's a lot of different kinds of equipment (shirts, pants, soup ladles, tennis rackets, shoes, socks, underwear...) that all affect different stats.
It's SO addictive. As far as I can tell, it was all made by one guy, and the translation is
Kawaiikochans-level hilarity. But the writing does have a great sense of humor, and there's a good amount of variety for a free game. I've seen a couple of realmoney options in the game, one for a pack of stat-up items and one for a pack of any-color modifiers, but I don't see these as being necessary at all.
My current character (easily shared from ingame):
http://i.imgur.com/0g41BjA.png