I moved from iPhone to an Android phone recently, and among dozens of iOS apps I've tried over couple years, some deserve to be mentioned. I include links where appropriate for convenience, but have no affiliation with any of the developers.
DayOne is a very nice journal application. I've used it to record what I've done each day, and then review entries weekly and copy into a different application. It served that purpose very well - the UI is clean, there's reminder feature, and swipe navigation between entries. It also supports sync via dropbox, so I could make entries on a phone and review on a tablet.
DocScan is an app to take a shot of a document with camera and convert it into PDF, as if produced by flatbed scanner. Its primary benefit is perspective correction that works very well. The files can obviously be sent by email, or added to cloud storage, though I found the number of required clicks a tad large. There's also some automatic conversion to black-and-white but it never produced results I've liked, so I always disabled it.
Pretty much every fitness app is about using accelerometer and GPS to track steps and runs, which is quite boring, but I've came across two kinds of different apps.
Runtastic Push Ups is a push ups trainer and tracker. You put it on the floor, and it uses proximity sensor to count your push ups. Or, you can press the screen with you nose. That sound a bizarre idea, but I found it a fun in practice. They also make two other apps - Pull Ups and Squats - that use accelerometer to track something other than steps.
There are also apps that flash and built-in camera to measure heart rate. Cardiio does exactly that, while Azumio Stress Check determines your stress level using heart rate variability. The latter can be quite entertaining.
In this saturated app category, I've picked Deep Relax and Self. The former has about 40 different sounds you can mix, and set an arbitrary timer. The latter is beautifully minimalist, maybe too simple for me.
I've tried a crazy number of those, and only a few were still installed after 5 minutes. Those that survived were Step Journal, Charge, TnS, Lumen Trails and rTracker, but none of them ended up actively used. In particular, rTracker, which is widely praised, ended up ugly app that could not draw sensible charts.
This mini-review is concluded by the single game app I found installed, called Music Tiles, where one clicks on black tiles that scroll from the top, producing sounds, and trying to go as far as possible. That's surprisingly addictive, especially on long-haul flights.