AquaMail - a better IMAP/POP3 email client

Flipboard - magazine-like news-reading app

Ad-free Android - adds a load of advertisers' domains to /etc/hosts (REQUIRES ROOT)

Andmade Share - a replacement “share” menu that allows for apps to be reordered as well as sharing one thing with multiple apps

Audio Manager Pro - lets you create audio profiles which can be scheduled, changing volume at set times, etc.

Barcode Scanner+ - pro version of the excellent barcode scanner app

BSPlayer Free - one of the best free video players

Chrome - Chrome! It's supposed to be fast, but isn't, in my experience

CloudPrint - print from Android apps

CMarks - access Chrome's synced bookmarks from Android

ConnectBot - excellent SSH client

CoolReader - one of the best e-readers I've found for random .txt/etc. files

DocsToGo - reasonable office app

Dolphin Browser - much better web browser than the stock one

Dolphin desktop toggle - toolbar button for Dolphin to quickly switch between mobile and desktop user-agent strings

Dolphin Wikipedia search - Wikipedia search button on Dolphin's toolbar

Dropbox - it's Dropbox…

ES File Explorer - very fully-featured file explorer, inc. SMB, WebDAV, FTP, etc.

Evernote - cloud-stored note-taking app

Eye-Fi - not much good if you don't have an Eye-Fi card in a digital camera, but very useful if you do!

Facebook - Facebook! Yay, I guess.

Firefox - Firefox on Android

GMail - GMail on Android

Google Authenticator - 2-factor auth for Google accounts

Google Calendar - Google Calendar app because the native Android calendar app doesn't integrate too well with Google Calendar

Google Music - Google Music

GPS Status - useful app for getting detailed information about GPS reception, etc.

Hoccer - “fun” way of transferring files between mobile devices; odd, but often also the easiest way…

IMDb - appified version of IMDb's website

Imgur Mushroom - weird name, but the best imgur uploader app I've yet found

inSSIDer - WiFi network analyser app

Jump Desktop - very good remote desktop (RDP/RDC & VNC) client

Kindle - Amazon's Kindle app

Last.fm - Last.fm client with radio streaming if you're a subscriber

LastPass for Dolphin - Dolphin browser integration for LastPass

LastPass - passable LastPass app. It's ugly, slow and a little frustrating to use, but better than having nothing at all…

Latitude Shortcuts - you're no longer able to launch Google Latitude directly from your apps list (it's part of Google Maps); this app restores that ability, which is invaluable for those of us frequently wanting to know where our friends are ;)

LinkedIn - LinkedIn app for those of us who use it

Google Maps - need I say more?

MemoWidget - lets you add a simple textual memo to your home screen

MyTracks - Google app to track your movements over a journey (e.g. walking, cycling, driving, etc.), then report back statistics and a plotted route

OkCupid - mobile app for OkCupid. Their developers need to take a read of the Android UI guidelines sometime, but mostly it's a nice enough app

Photo Editor - a very functional photo editor app. On older versions of Android it can't allocate enough memory to work with larger images (Android 2.3 combined with 12 megapixel images failed), but later Android versions are fine

Pocket (aka ReadItLater) - save articles to your Pocket in order to have it download them for offline reading as well as keeping a handy list of things that you need to read at some point

Popcorn Hour Remote - network-based remote control for Syabas Popcorn Hour media devices

ROM Manager - useful ROM management utility for rooted devices (REQUIRES ROOT)

Samba Filesharing - sets up a Samba server on your device in order to share your SD card over the network (REQUIRES ROOT)

SpeedTest.net - mobile version of the speedtest.net service; useful for testing your 3G speeds

SugarSync - client for people who use SugarSync's syncing/backup service

TED Videos - watch TED talk videos all neatly compiled into one place

Tesco - Tesco's app does various things, and no longer requires the overreaching “see list of running apps” permission!

TitaniumBackup - excellent (if confusing) backup utility that's invaluable if you flash new ROMs onto your device (REQUIRES ROOT - I think)

Twitter - Twitter app, duh

Watchdog Lite - keeps an eye on the CPU usage of apps and will alert you when an app starts using too much CPU (which Facebook has a habit of doing, totally out of the blue)

YouTube - YouTube app

Compass - use your device's internal digital compass to display a graphical compass on the screen

FileExpert - file manager utility

FlickFolio - Flickr app. I think I installed this primarily for uploading stuff from my tablet

Google Drive - used to be Google Docs; still does much the same thing

Google Goggles - fun little Google app

Google Books - because we don't have enough ebook apps

Google Reader

Google Plus

My Xbox LIVE - Xbox app because Smartglass doesn't work on tablets yet :(

Quasseldroid - Android client for Quassel

ScreenshotIt - screenshot-taking app for devices which don't make it easy to do that normally (requires root for most devices)

Google Sky Map - Google Maps for the sky!

3G Watchdog - keeps track of 3G data usage, with a nice widget and such (had more utility before Android had similar functionality built-in, but the widget is still nice)

Caffeine - keeps the screen on and/or set to don't-dim-on-idle when connected to a power source (e.g. a desk/car dock)

CSipSimple - a reasonable SIP client

DeskSMS - allows sending/reading of SMS messages through a Web interface (or via TabletSMS on an Android tablet; see below). NOTE: Still seems to have a bug where sent messages longer than 512 characters are cut off. Keep messages short-ish!

Foursquare - Foursquare client!

SMS Backup+ - automatically syncs all sent/received SMS messages to your GMail account for ease of searching and backing up

SMS Blocker - surprisingly-good spam SMS blocking app

SMS Popup - displays incoming SMS messages in a pop-up dialog with options to quick-reply, read the message aloud using TTS, etc., and can be configured to have different notifications for each contact

Xbox Live Smartglass - Smartglass is Microsoft's new Xbox companion tech. Primarily I use it to let my phone act as a remote control for my Xbox, which is especially useful for entering text, where I get to use my phone's onscreen keyboard instead of the Xbox's

Dolphin for Pad - a tablet version of Dolphin browser. Unfortunately it's very buggy, crashes frequently, and often switches tabs for no obvious reason

DroidEdit - code editor for Android

Maxthon Tablet Browser - a very fast Web browser for tablets, albeit with a slightly odd UI. Much, much faster than Dolphin

Evernote Skitch - sketching app for Evernote

Tablet CPU Usage Monitor - monitors CPU usage of tablets, placing icons in the notification bar displaying CPU/RAM usage

TabletSMS - the Android tablet client for DeskSMS to allow sending SMS messages from your Android tablet via your Android phone (thus using my Asus Transformer's hardware keyboard to type messages)

Angry Birds

Angry Birds Rio

Angry Birds Seasons

Angry Birds Space

Apparatus

Blow Up

Doodle Jump

Draw Something

Fruit Ninja

Jewels

Kinectimals

OnLive

Rayman Jungle Run

SliceIt

Spaghetti and Marshmallows