I recently entered into the world of rooting, my current being out of warranty and with no forthcoming future updates, rooting giving me access to new more features on my Android phone and even allowing to install custom ROM's. Because of this I am have made a few additions to my list of best apps. While this is not a list of all the apps on my phone, I consider this my must have apps.
A note, I did not include in my list HTC proprietary apps installed on my phone since they are not available unless you buy an HTC phone. This is why you will find some gaps on my list. I use my HTC Proprietary apps for the dialer, SMS, calendar and Twitter.
1. Swiftkey 3 Keyboard (Paid). Literally the app I use most. Since I bought this app a year ago, I have received the next two major revision free. It is the best Android keyboard available. Swiftkey, combines excellent word prediction with a highly customizable keyboard. It learns your habits and can even predicts whole words for you before you type the first letter of the next word. You can do more than just change the theme, you can change the size of the keys. Give it a try.
2. Gmail (Pre-installed/Free). For Gmail users, the best email app is Google Gmail app. It basically gives you the same degree of functionality as if you were on your desktop.
3. Dropbox (Free). Drop box keeps my files in synced between my desktop replacement and ultraportable laptop and allows my phone to access the files synced. You now have Google Drive for Android which gives you more free space. But Dropox only gives you 2 GB of web storage free, but can work your way up to 20 GB via referrals and tasks. Sign up through this link, and you start out with 2.5 GB.
4. Tasks (Paid). One thing Google does not have is a Google Tasks client for Android. There are many apps in the Play Store that sync with Google Tasks. Task by Team Tasks does so elegantly. Some of the other apps try to do too much. It looks and feels like an official Google ICS app. Has excellent widgets too.
5. Pocket (formerly ReadItLater)(Free). Android's Instapaper. A great way to save links to view later on your phone or other device. Pocket will sync your unread items list and also formats articles in a phone or tablet friendly format. Instapaper is not on Android too, but Pocket is the better app.
8. Screen Lock (Free/Donate). I use this to lock my phone using the capacitative search button. This saves some wear on tear on my power button. Free app is fully functional, but I got the donate version. Wanted to support the developer so that we see more great stuff from him.
9. Evernote (Free). Android has many excellent note taking apps with cloud storage, but with new Android specific interface Evernote is the best.
10. Viber (Free). Viber is a VOIP and instant messaging app. Unlike Skype, Viber was born on mobile so it is is much more resource efficient than Skype. The app even supports push notifications on Android. Available on other platforms, this is a good way to communicate with friends for free. Viber customer support is also second to none.
11. ezPDF Reader (Paid). The key feature of this PDF reader is text reflow. It can wrap the text on PDF documents so they fit the boundaries of the screen.
12. Wopnersoft Unit Converter (Free). There are plenty of good unit converter in the Android Market. This one works well and is add free.
For the user with a rooted phone -
1. SuperSU (Free/Paid) - Rooting your phone will not be enough to give you access to the advance Android management tools. You will need super user access management tool. This app will also give you a warning whenever a app tries to obtain super user access. I use SuperSU by CHAINFIRE right now, but have also used SuperUser by CHAINSDD. They are both fine apps. Free version of both are available with paid versions which are really donate versions, since while they give you access to additional features, the free versions contain everything you need.
2. BusyBox (Free/Paid) - Most apps which take advantage of root access applications needs BusyBoxyo work. In order to install BuxyBox on your phone you need a BusyBox installer. This app install BusyBox on your phone. After you install the app, run it and install BusyBox.
3. Root Explorer (Paid) - One thing you will need once you root your phone is a file manager to poke around your SD card, or even your internal memory and system partition. There are plenty free apps that will do the job, but I recommend Root Explorer. It is more than just a tool to look around the your phone, but also includes a SQLite database viewer, text editor, a tool to create and extract zip or tar/gzip files, extract rar archives, a tool to execute scripts, an APK binary XML viewer, a tool to create symbolic link, an MD5 checker and more.
4. Titanium Backup (Free/Paid) - The free version of this app allows you to back-up files and data on your phone. It is fully functional. You will need this if you plan to install different ROM (versions of Android) on your phone so that you can get up and running quickly. It is also useful to back-up apps, so in case you do not like the new version of an app, you can go back to the previous version by using your own back-ups. The paid version allows you to back-up your files online either to Google Drive, Dropbox or Box. The paid goes beyond a back-up app and is a fully powered app manager with the ability to freeze apps (render them inactive without uninstalling, uninstalling system apps (those built into the ROM that some call bloat), break market links so that if you decide to stay with an older version of an app, you wont be notified of updates. It can restore those links too. Beyond those it has more feature I have no explorer myself, so I will stop here.
5. 4EXT Recovery (Free/Paid). If you want to install new ROM's, Radio's or themes on your phone, you will need a recovery program. My favorite is 4EXT Recovery. You can install 4EXT Recovery using a free updater. You need to boot into recovery to access 4EXT Recovery. After selecting Recovery with your volume and power buttons, it has a nice touchscreen interface. You also get a paid version, 4EXT Recovery, which allows you pre-set from the phone UI itself what you want it to do, so the phone will reboot itself into recovery, 4EXT will execute what you asked it to, and the phone will reboot back into Android. In addition, 4EXT makes all this safe with a built in md5sum checker.