I can't give you any S3 specific ideas since I don't have the phone (not for lack of wanting, mind you).
Here are some apps that I use heavily on my phone/tablet, both running ICS.
Tasker - amazingly powerful way to automate settings, actions, etc. The sky is the limit here though it does have a little bit of a learning curve. There are lots of ideas on the product website/wiki.
PureCalendar widget - I have a calendar widget running fullscreen on one home screen that shows the calendars I choose (in my case, just my work and primary google account). It integrates well with tasker (for example: my ringer gets automatically silented when I am in a meeting). Speaking of which, if your workplace uses Exchange 2007/2010 and has ActiveSync enabled... the integration with Android is excellent. One downside if integrating your work exchange account though: you have to use a PIN or Password to unlock your phone, not just a swype pattern.
Google Authenticator - combined with 2 factor authentication on my google account, this is a must for any apps that wan to integrate with my google login.
SetCPU - requires root. I use it to replace the default CPU governor to get better battery life.
Any.DO - great todo manager, also has a chrome plugin
Out of Milk - decent shopping list type app. Not amazing but I haven't found anything better.
Chrome or Dolphin - Chrome works great on my tablet but seems slow on my phone. Dolphin is a great alternative and is very zippy. My only complaint with Dolphin is that it doesn't support text highlighting nearly as well and doesn't have the nice phone type integrations such as recognizing when I long press a phone number.
Google Reader - RSS reader. 'nuff said
ScoreMobile - easily the best sports score app
Talk - I use this more than SMS. I tried several different apps in an effort to include Skype or AIM accounts in a single messaging app, but frankly its the protocols themselves that result in either crappy battery performance or crappy responsiveness/multiple notifications so now I just use Talk and resort to only communicating on AIM or Skype through the native applications when I choose to run them.
Random Apps that I have found useful occasionally
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IMDb
Yelp
Dropbox - auto picture upload is great, but I think the S3 might come with something comparable built in
Zedge - ringtone library
Redmine
Subdroid
GreatClips
Netflix
Manger - ventrilo client
Dice - its a dice app, so simple but so useful
Timer
Pandora
QRDroid
TVListings
IPcam Viewer - if you have any IP cameras
Navigation/Maps - pretty solid google apps, haven't had a reason to try to look at alternatives
Superuser - superuser manager (requires root)
WiFi Tether - better than the native tethering
BubbleUPnP - best UPnP capable browser/player that I've found so far, the S3 might have something integrated
MxPlayer - hardware aware video player
ConnectBot - ssh client
Games
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World of Goo
Game Dev Story
Androminion
Jelly Defense
Sentinel 3
Angry Birds
Apps I recommend avoiding
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Facebook - battery hog, current HTML5 version has crappy performance
Lattitude - battery hog
Skype (running all the time) - very load intensive. Fine for loading when you want it and then exiting but still kinda annoying. Talk has support for video chat now, which I prefer.
IRC clients of any kind - trying to use IRC in any way over a mobile connection is just silly. The protocol is to blame, not the clients.
