Okay, let's go through my phone (iOS :| ) to see whats up:
FreeGraCalc (Graph Calculator) - I don't use the graphing features much, but it has the power to do that. I mostly just like how they made it simple, yet powerful. For free.
Evernote - It syncs everything you write to the internet. You can add words and pictures and even videos. Also have a Windows version which syncs to the same account. Good for random thoughts that come to your mind, or planning things.
Mathway - If you get lazy when doing homework or something. It answers pretty much any math problem for free, and if you pay, shows you the work. Might be good for checking your answer as well.
PhotoShop Express - Better than the standard camera app and instagram for editing photos on the go. If you want to add filters, flip/rotate, etc.
OOKLA Speed Test - It tests your internet speed. Works fine.
remoteMouse - I use this when I need to hook up my PC to a TV. It can control mouse using the touch screen and type using the touch keyboard as long as they use the same network. Way easier than trying to use a mouse and keyboard in your lap and tripping everyone just to watch a movie. Only downside is you have to go into the Desktop for it to activate, so have the keyboard with you at least to log in.
Google Maps - Arguably better than Apple's Maps app.
Steam - No games, but if you wanna look at games or stuff on steam it's alright.
Imgupr - Easily upload to Imgur.
Sybu Kodi - Basically a remote for Kodi(XBMC) frontend on an HTPC.
Modizer - Download and play those Amiga chiptunes and such from anywhere on the web.
NesMusic - If you like chiptunes, you can easily add a .nsf file from your favorite NES games. Then listen freely. Paid can get other systems like SMS Genesis/MD SNES etc.
SleepyTime - Basically sleepyti.me, but on your phone, and with a better interface.
Habitica - A habit motivation app. It is like an RPG, you have HP and if you do bad habits you get hurt. Do good habits and you get HP and gold. Use gold to buy more armor and get steeds. It's alright.
Dropbox - You know what this is. Works just as well as on PC.
Finish - If you need to finish something by a dead line, this app is good. It will bug you sometimes about it and when it's urgent. Plus you get free coupons (most are redundant, but still.)
PIA VPN - It's alright, sometimes it doesn't work and the internet stops working. Just reset it. Easy to go in and out of the VPN. Need to buy the VPN service first, it's not free.
MediaConnect - If you have a DLNA media server, you can connect and listen to/watch it, for FREE! Supports most formats of music including lossless ones. I don't use it for video though, so I'm not sure about those.
IRC Cloud - Chat using IRC. Self explanatory.
Alien Blue - A reddit client. I actually liked iAlien better, except you can't see what you are typing during a text post.