Here is what I'm using (as of July 4, 2013), on my Android phone (a Motorola Droid 4). They're all free unless otherwise noted.
handyCalc: A calculator designed by a geek. Built in unit and currency conversion, can define functions, graph, and more.
OI Safe: Open source password safe.
VX ConnectBot: Recommended if you need a good SSH client. It's a fork of ConnectBot, which is also good. I'm only using VX ConnectBot because standard ConnectBot has problems with my keyboard.
Barcode Scanner: Fast and just works. Reads QR codes, but can also look up UPC and other barcodes.
Kaiten Mail: A solid email client. Available with ads, or in a paid version. If you don't want to pay and don't like ads, consider K-9 Mail, which is Kaiten is based on.
OI File Manager: You'll occasionally need to root around in the files on your SD card, and this is good at the job.
Droidlight LED Flashlight: For Motorola Droids only, turns on the camera flash for use as a flashlight.
OI Flashlight: A straighforward app to turn your screen all white to use it as a crude flashlight.
OI Shopping: Serviceable shopping list app. I'm not actually all that fond of it, but I haven't found a better one yet.
Google Authenticator: Two-factor authentication is awesome!
Quiet Time: Turns off the ringer for a set amount of time. Useful when out at the movies.
Stopewatch & Timer: Exactly what it claims to be. Can't set the alarm or volume for the timer, but it's good enough for my needs.
SketchBook Mobile: Pretty good drawing program, sort of Photoshop ultra-light.
Various services I use: The Android clients are all pretty good these days. So get the clients for whichever social networks or cloud services you use and call it a day. At the moment I have and use: Evernote, Dropbox, Netflix, Facebook, Google Plus, Twitter, Newsblur, Gen Con Mobile
Shortyz Crosswords: Very good crossword puzzle app that pulls puzzles from a variety of online sources, including several newspapers.