Here's some that I get a lot of use out of. Your mileage may vary:
QuickOffice Pro - I use this on the Kindle Fire more (I actually do a lot of writing on that one), but it's worth mentioning. Word, Excel, Powerpoint, PDF functionality. The Pro version lets you create docs.
Amazon Store - It's actually got a pretty nice mobile interface. Fun to browse, and deals crop up from time to time.
Audible - Audiobooks, if you're interested in that sort of thing.
Gmail Authenticator - If you use gmail, I'd definitely recommend using it.
CalenGoo - Solid calendar app that seems to do a much better job of synching with Outlook/Google Calendar than the default junk. Good widgets.
ES File Explorer - I always find that a decent file explorer is a must-have. There are a bunch, though I've only used ES. It does the job well.
Evernote - Notes with synch across all devices.
IMDb - I use this more often than I'd like to admit.
Mint.com - Personal finance. I gather that they're part of Quiken (sp?) now. Used this until it stopped communicating properly with Bank of America, but it was great while it lasted.
ReadItLater - You can save website pages to it to read offline. Synchs across all devices.
Sketchbook Mobile - AutoDesk's sketching program. I use this much more on my Kindle Fire, but it might be worth a look regardless. Wish they'd add a blend tool.
Skype - It's Skype.
Shush! - With this, you can set your ringer to silent/vibrate, and it'll return back to a set volume after a set amount of time. Useful for setting the phone to vibrate during work hours, or during a movie.
Astrid - Decent task manager. There's a million out there, though.
Banking App - If yours has one, get it. I use BoA's a veritable ton.
Google Translate - It's so-so. I use it once in a blue moon.
SlideIT keyboard - It's probably overpriced now. I grabbed it when it was on sale for $.10. Lets you slide from key to key, instead of typing each one individually. Recognition/prediction is decent, which is probably about as good as you can expect. Easy setup.
Pulse - News. Really nice interface, really customizable. You pick what sources you want on there. It came default on the Fire, so I dunno if it's a paid app, or how well it works on phones. But if you can, I'd take a look at it.
As far as battery usage goes: The biggest advice I can give is make sure that your GPS is off if you're not using it. That thing will eat your battery fast.
Edited, Jan 17th 2012 11:47am by Eske