All of the following apps are my favorite apps, and they're free. (Some accept donations and deserve them!) Listed in alphabetical order but asterisks denote apps which I personally consider must-haves. "ROOT-ONLY" means that you must be rooted to use that app.
Android Speedometer: Speedometer with peak speed, etc. (Fun for skiing, biking, etc.)
AnyMemo: Create flash-cards (for studying) or access pre-made flash card sets (e.g. Spanish 101).
Avast Antitheft*: Remotely locate, disable, & wipe data from a lost or stolen phone.
DishAnywhere*: If you have a Dish Network satellite receiver, watch anything (live or recorded) from anywhere in the world-- including inside your own house.
ES File Explorer*: Manage/move/rename files & folders on your SD-card (and, if rooted, even your internal memory). Transfer files to/from PC via WiFi, etc. Access all of your cloud drives, like Box.net, OneDrive, Google Drive, Dropbox, et al, from one app. Does much more too. A must-have Swiss army knife.
Gas Buddy*: Find the cheapest gas near you and get GPS-directions/navigation to the station.
Google Goggles: Scans bar codes and even real objects (like a coffeemaker) to help you price-shop, etc.
Graphing Calculator: Has 3 modes: Basic, Scientific, & Graphing.
GrooveIP: Free phone calls over WiFi.
IP Cam Viewer Lite: View traffic cameras or home webcams.
Link2SD*: ROOT-ONLY. Effectively & selectively expand your internal storage by moving apps/data to SD-card (saving more memory than Android's built-in app2SD functionality).
Llama*: "Location-based" (if-then) actions. It determines your location without battery-draining-GPS (from cell phone tower data) and automates "tasks" based on (1) your location (e.g. go silent when at church) and/or (2) your connection/proximity to other things (e.g. if car-stereo Bluetooth detected and power adapter connected then unlock screen, go full brightness & volume, launch Waze GPS, & launch Pandora music).
MediaMonkey*: My favorite music & video player app. If used with the excellent, free media player/manager MediaMonkey for Windows, MediaMonkey for Android can synchronize all (or if you like, some folders of) music files & tags, optionally album art, video, audiobooks, podcasts, & playlists from PC (including WMA files & metadata). As far as I know, it's the only truly complete desktop-to-device solution for Android (like iTunes-desktop-to-iOS-device, but more powerful). The Android app has an equalizer that even works on WMA files (a rarity). It can play songs in track # order, not just alphabetical order (which is also very rare despite being critical for playing albums where track # order matters, like Pink Floyd's The Wall).
MediaMonkey Remote*: Let's you remotely control your PC's MediaMonkey player via WiFi, including library & playlist selection, existing playlist viewing and editing/rearranging, volume, play/pause/FF/rewind/next/last/skip-forward-a-few-seconds/skip-back-a-few-seconds. Nice if you use your PC as a music and/or video player connected to your living-room stereo and/or TV. Again, part of a complete solution.
MixZing: If you don't want a complete, synchronizing desktop-to-phone media solution, MixZing is my favorite player app because it's a Swiss Army knife that plays music, video, Internet-radio, and has one-click access to lyrics & artist-bios. It has an equalizer and can play by folder.
MortPlayer Audio Books: Audio book player, with bookmarks, skip-back, etc.
MXPlayer: Video player app that can handle almost any file type.
Pandora*: Free music streaming. Create your own "station" or use a preset station.
Playlist Backup: Backup playlists or import M3U playlists (probably from your PC). Nice if you don't use a syncing media player like MediaMonkey.
Playlist Designer: Easily make playlists based specific songs, artists, folders, genre, etc. Nice if you don't import playlists from your PC with something like MediaMonkey or Playlist Backup.
OurGroceries*: Create & share (probably with family members) shopping lists, packing lists, to-do lists, benchmark-price lists, etc. Updated in real-time so, e.g., 2 people can simultaneously shop from either end of a grocery store or pack for a weekend trip. Items are crossed off the list and placed at the bottom (in strike-through font), so they can easily be re-added so that you don't have to re-type (or re-voice-to-text-add) items, or always look up your printer's ink-cartridge part number. Lists can be EASILY managed from any web browser also.
Quick Settings- by Halfbit*: Quickly access practically all of your phone's settings.
Scanner radio: Police scanner.
Screenshot: ROOT-ONLY. Take screenshots of your phone's screen.
Silent Time*: Set times when phone should not ring, except optionally for alarms & calls from specific people.
SoundHound*: Listens/identifies song name & artist for music playing in the room.
Titanium Backup*: ROOT-ONLY. Let's you back-up your apps, data, & settings.
TeamViewer: Let's you access & control your PC, Mac, or Linux PC.
Torque: If you have a ~$16 Bluetooth OBDII dongle for your car's OBDII diagnostic port, it can read diagnostic codes, view engine data, and help you save money by fixing your own car (or at least be knowledgeable about the problem when you take it to a shop).
Ultimate Stopwatch*: Stopwatch & countdown timer.
Ulysse Gizmos*: GPS coordinates, altitude, compass, bubble-level, speedometer, GPS satellite tracker, etc.
Utter!*: Like "Siri", it lets you control your phone with voice commands, like "Launch Pandora", "Kill Pandora", "Listen to Rolling Stones, Some Girls", or "Navigate to Radio Shack". It can be triggered by selecting the app's icon or typically by long-pressing the phone-off-hook button on most Bluetooth-enabled car stereos (or headsets) making it a nice way to control your phone, hands-free and heads-up, while driving. (But be careful!)
WatchESPN: Watch ESPN's live and previously broadcast events.
Waze*: Maps & turn-by-turn GPS (with real-time re-routing based on traffic), warnings for cops/radar, hazards, etc. The traffic & hazard awareness makes it better than Google Navigation, IMHO.
Where's My Droid*: EASILY, QUICKLY, locate, track, or blare your lost phone with by sending user-specified text messages, like "Find my cell" (and it will text back a Google Maps link to its location) or "Blare my cell" (and it will blare an alarm sound at full-volume).
WinRemote: Use your phone as a PC mouse (which is nice for giving Powerpoint presentations).
WPS Office (formerly Kingsoft Office)*: Read, create, & edit most document types (e.g. Word, Excel, PPT, PDF, etc.)
