I'm running a Samsung Galaxy S2 that I've rooted. Here's my most commonly used applications, all of which I recommend:
* Smart Keyboard Pro: https://market.android.com/details?id=net.cdeguet.smartkeybo... -- Allows me to use a Dvorak keyboard on the phone, but even if you don't use Dvorak it's still a great purchase. It has a lot of options to make your typing more efficient that the stock keyboard doesn't have.
* The Kindle app to read books. Now I mostly use it to read .mobi files I've downloaded instead of buying things at the Amazon store, for which it works brilliantly. I wish it allowed copy/pasting of text though.
Google Reader for RSS/Atom
* GMail application for GMail.
* The stock Mail application for work IMAP, although I'll test out K-9 mentioned in this thread
* ColorNote for taking notes / shopping lists.
* live logcat for the novelty of having a l33t boot sequence (and getting rid of that annoying Samsung boot-up sound): https://market.android.com/details?id=eu.chainfire.livelogca...
* Root Explorer to manage root-only files.
* Skype to make video calls
* ConnectBot to ssh from the phone, although I haven't found out how to enable landscape mode in it, which is annoying.
* Listen for podcasts, although I haven't found out how to make it auto-download say the last 3 podcasts of everything I subscribe to.
* BBC News and Al Jazeera video for some world news
* Gh4a to browse GitHub
* AdFree to block spam
* Screenshot to take screenshots
* WolframAlpha for any sort of conversion / math queries.
* A PushUp / SitUp / Squats app (all separate apps) for managed stationary exercise. Although I'd like to find something that I could turn on and would give me instructions for a general N-minute workout that could be done at home.
Things I wish I didn't have to use, or don't know how to do, but haven't found out how to replace / do:
* The default Samsung camera application. There's no way to turn off the shutter noise on it, and it'll plain refuse to take pictures if the battery is low.
* The default task switcher, which often doesn't list the application I've just switched from (even though it's still open)
* The default system clipboard which doesn't seem to have a history / search feature like Emacs's kill ring.
* Having the default Samsung apps installed on the phone. I tried to use Titanium Backup PRO but it didn't work for removing or freezing them.
* Having something like Gesture search which searches across the entire phone but doesn't require me to draw on the screen, I just want to type a query.
* Sometimes the phone will go into some sort of voice command mode. I haven't found out how to disable it.
Lastly I plan to write an application that hooks into the "Share" handler for text/plain content and extracts the content of the clipboard, scans it for a URL, and pastes that to the clipboard.
I've found that quite a few apps have a "Share" functionality, but no "copy the URL of what I'm looking at to the clipboard", and I haven't found any existing app that works around this.