The top two apps I use is twitter and the STEAM app beta. Mango is cool if you are into Manga and ASTRO to configure files when you don't want to plug into a computer.
Google Voice. Set up a free google voice number, and give that to everyone in your contacts list - just say you got a new number. It's a bit painful of a transition, but here's what you get:
Free Visual Voicemail
Free Text Messages (they are sent over data) - I have disabled SMS with my carrier (Verizon) and rely solely on GVoice
Access your calls and texts from any internet enabled device - and Chrome even has an extension so you can send and receive texts from the browser.
Set up your GVoice number to ring to other phones (work, mobile, home, etc)
Personalized greetings per contact
Great integration into Android
The cons are not many, but are pretty annoying:
Essentially changing your number and making everyone use the GVoice number
Can't send or receive picture messages - this is the worst. Messages essentially disappear if someone sends a picture message to you - you'll never know you even got it. I (assume) I don't get many of these to begin with, so the cost here for me is not that great, but your mileage may vary.
I really enjoy the Hanging with Friends game. It is a lot of fun.
I also like the GoogleMaps app because it is really accurate and helps me if I am lost!
Soundhound is one of my favorite apps. I no longer have to live in ignorance of what a particular song I'm hearing is called. Tap the "Listen" widget and, after a moment, the app will tell me the title and artist.
Zedge. It's free ringtones and wallpaper. It has a great search function so you can search for anything you like; video games, tv show, favorite singer, whatever. Plus it's easy to use!
There's a myriad of NES/SNES/Genesis/GBA emulators out there, my personal favorite is Tiger NES, (also use a RAR extractor which is also on the marketplace for free "ArchiDroid") and download roms from Emuparadise.org. pretty simple once you understand it.
1: JuiceDefender. It's free, simple, and it nearly doubled my battery life. No more dying in the middle of the day and having to be charged, no sir.
2: The Humble Android Bundle. Bunch of quality indie games for whatever you want to pay for them.
3: Gameboid/Snes9xEX. Once you have these, you can play any gameboy advance or super nintendo game you can get your grubby hands on.
Aside from social media apps, I find that Google Finance and DroidLight are the most useful, as I frequently need a flashlight, and checking on my investments.
I use my banks app as well as a crossfit travel app. They are two of my most used and best applied apps. The other I use is a free checkbook app that keeps a running tab of all of my receipts, checks used, and bills paid, etc. It's called cashflow.