The apps I use the most would be;
Classic notes and app box. There really doesn't seem to be much it doesn't do. There are so many functions it's insane. I mostly use it as a notebook and task manager, and it does scheduled reminders. Definitely one to get.
Dropbox - lets me transfer files from the S3 to any other device I have. And I mean any.
Flipbook - awesome news reader.
I am testing Juice Defender - I've been doing that for a week now and the results are uneven. I plan to run it for a few more weeks, collecting battery data as I go with it alternately enabled and disabled to see if I can get some good stats out of it. So far the results seem to say that their claims are overblown, which is no surprise. But if it works it will be worthwhile.
HD Widgets - better imho than Beautiful Widgets - very nice clocks and weather widgets, plus some useful switch widgets.
Brightness Level widget - a simple, 1x1 widget that, when you tap it, gives you a choice of 0, 25, 50, 75 and 100% brightness, plus can toggle auto. Very nice and free. There is a similar widget in HD widgets which is prettier, but it doesn't work as well.
Kindle of course. I have a lot of Kindle books and the S3 makes a great reader. Plus I carry a lot of UL standards in PDF form and the Kindle reader works well with them.
Wolfram Alpha. Everyone should have Wolfram Alpha. It's just awesome. It's what Siri uses to answer a lot of questions. But it seems to know everything and it can solve any problem or equation you throw at it. It's not free, but it is very worth it.
These are the apps I use on a daily basis. I left out electronics apps, but there is probably a limited audience for those. Tesla is about the best though.