(All of these are apps I have personally used, with the exception of the automation apps, IRC Radio, Greenify, CloudMagic, Helium, and Wigle.)
AK Notepad / Notes
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Automation: IFTTT, Tasker, Condi, AutomateIt, Automagic, Trigger
Avast Mobile Security (I don't really need the virus protection, but it saved my skin when I forgot my unlock PIN code once!)
Bubble (It's a bubble level!)
CamCard Free (Snap pics of business cards, extract info from them using OCR, and keep them in a digital cardholder. Less to keep track of!)
Compass (or there's “Compass & Level”)
ConnectBot (SSH client)
Duolingo
Drippler
ES File Explorer
GasBuddy
GitHub
Google Keep
Google Translate
Google Wallet
Greenify
HealthTap
Hotel Tonight
Idea Growr
Lift
K-9 Mail (old choice. Newer choice: Boxer. Third choice: CloudMagic)
MapFactor: GPS Navigation (I use this instead of Google Maps because my tablet doesn't have 3G/4G connectivity and it allows me to download offline road maps.)
Musical Piano
Nova Launcher
PaperKarma (Snap pics of your junk mail and it will attempt to unsubscribe you automagically.)
PocketCloud Remote RDP/ VNC
Quadrant Standard Edition (a hardware benchmarking tool)
RealCalc Scientific Calculator
RedLaser
Sakasa Kotoba (a simple audio reverser. play something into it, press a button, and listen to it backwards.)
Simple (an online bank account app thingy)
SleepBot
Speedtest.net
TED
Terminal Emulator for Android
Titanium Backup (Newer choice: Helium)
touchqode
urWell
Wifi Analyzer
WiFi Keyboard
Wigle Wifi Wardriving
Wordpress :)