In an effort to reclaim memory on my Xperia Arc, I recently did a hard reset on my phone and went through the trouble of re-installing only my most essential Android apps. Here in alphabetical order are the ones that I re-installed immediately.
ACast - By far my most used app. It's a "podcatcher" app and I listen to podcasts more than just about anything else. ACast is not user friendly, but it is free and it gets the job done. It has some pretty important features like a home screen widget and limits on when to download episodes (e.g. "only while charging on WiFi"). Most importantly, it works well with my favorite Bluetooth headset (Sony Ericsson MW600). If it was a little cheaper I might actually buy the "pro" version just to eliminate the ads, but for $4 I'll put up with them.
Advanced Task Killer - This is simply the quickest way to kill everything running on your phone. When it feels like your phone is sluggish, ATK will quickly bring it back up to snuff.
ASTRO File Manager - You simply gotta have a file manager and Astro is a fine one. I also like Adao but I'm using Astro right now. Astro also has a task manager function that makes Advanced Task Killer a little redundant, but there are too many clicks necessary with Astro. ATK is more convenient.
Bambuser - A great app/service to stream live video from a mobile device to the web. It's the easiest way for my parents to see live video of my kid without making them jump through a bunch of techy steps (download/install Skype, create an account, add me, etc.). My Bambuser live broadcasts can be seen at http://www.williamfrantz.com/p/live.html.
Blogger - Of course this blog is hosted by Google's Blogger service so having the mobile app is a handy way to keep my blog up to date while I'm out and about.
CamScanner - Essential! This does an excellent job of turning photos into PDFs. It's very handy for capturing receipts or other important documents. It's better than simply taking a photo because it has powerful cropping options and does a high contrast pass to clean up the images. Must have!
Evernote - Can't live without this either. I have Evernote installed on every device and computer I own. It keeps all my important notes backed up in the cloud and easily searchable from just about anywhere. I use Evernote daily.
Facebook - 'nuff said.
LinkedIn - Since I work in business development, my Linked In profile is probably more important than my Facebook account.
Mute-O-Matic - I actually bought this one. Simple app that automatically mutes my phone during meetings. It also allows for exceptions so calls from my wife always come through no matter what is on my calendar.
Pageonce - Fantastic app/service that aggregates all my bank accounts and credit cards into an easy to read dashboard. Also reminds me when I have bills coming due. This app has probably saved me serious money in overdraft/late fees.
Redfin Real Estate - If you are even thinking about buying a new house, this app/website is like crack. I pity the cities that do not have Redfin service but in San Diego, it is absolutely the most useful house shopping tool around.
SwiftKey X Keyboard - I bought it! You can keep your Swype and Siri. SwiftKey is hands down the best way to enter data on a smartphone. It does a fantastic job of predicting what I'm going to type next before I push a single key.
Twitter - 'nuff said.
There you have it. Of course I also use a lot of the standard apps too like Google Maps although I don't bother with the gMail app. Instead I use the standard Mail app to fetch both my gMail and Exchange mail so it can present them both to me in a unified inbox.
The only apps I've bought on this list are Mute-O-Matic and SwiftKey. I think ACast might be the next thing I'd pay for. All my other purchases are basically games/books for my kid.