Similar to my Windows tools post, this one focuses on tools that make life a lot easier - on Android - or help you pinpoint and troubleshoot problems.
You can find files, pack/unpack archives, change permissions, move stuff around, handle multiple tabs and work with SMB and FTP shares, Google Drive, Dropbox.
There might be some free alternatives (ES File Explorer, Root Browser) around but at that time (~2012) there weren't many tools like this..
You can back up an almost complete state of your phone/tablet and restore it to another device.
The backups can be shared between accounts, so you can restore a purchased program with complete settings on another account/device.
I run a weekly backup and just dump the complete contents of the internal and external sd cards to the NAS. Later in time I delete obsolete backups, but they are really handy when things go wrong or if/when you lose your phone.
Hacker's keyboard helps though.
Hacker's Keyboard - add arrows, modifiers and function keys to the keyboard
Alarm Clock xTreme - if you have trouble waking up this is for you
Pocket - share web pages between devices. Can cache them for offline viewing
RealCalc - a decent scientific calculator
SecondScreen - adjust HDMI output parameters: resolution, overscan, DPI, ...
Trickster Mod - play with max/min CPU frequencies and schedulers, see actual sleep time
Google Translate - the picture translate mode is really useful, when it works. Real-world scenario:
VpnRoot - can create a VPN connection without enforcing security on your phone
QuickDic - an offline dictionary, useful when abroad with no internet connection.