I'm still pretty new to Android but here are some apps I've found useful or find myself using regularly:
Grooveshark. The actual service does carry a $9/month fee (1 month free, I believe) but for me it is well worth it. Grooveshark has 50 times more of the genres and artists I listen to than Pandora.
1 Click Cleaner. While it's truly not a '1 click' solution, it seems to be doing a pretty good job for me in terms of tracking down and eliminating cache from just about any app on my phone. I believe there's also a pay-version for rooted phones but this seems to work fine for my stock one.
Skyfire Web Browser. I've tried the different flavors of Dolphin, Opera Mini, Miren and of course the factory browser but I like Skyfire the best. It's fast, plays most embedded/inline videos and reliable. I really don't care about all the extra 'social' stuff it does, honestly I wish they'd put out a version without all of it, but it's still my top browser choice.
Quick Settings. I'm not sure what sort of 'trickery' it uses but I do know that thanks to Quick Settings, I can now use my Optimus S as a wi-fi hotspot without incurring any additional fees from my carrier. In addition to this (it's really all I use it for, honestly) it also has quick on/off's for GPS, Volume and Ringer, Bluetooth, Wi-Fi and Mobile Data.
TweetDeck. I'm not a big personal social media user however I do run the Facebook/Twitter pages for the company I work for so something like this is imperative. Tweetdeck for Android has been just as reliable, feature-rich and easy to use as the desktop version. Now I can actually go out for lunch and still schedule my tweets for the week.
Google Voice. Google Voice is pretty much a necessity. Now thanks to having an Android phone I can keep tabs on my texts and calls either via the Chrome extension or anywhere with the mobile app. I didn't really like how it tried to virtually 'take over' (inquiring as to whether I wish to use Google Voice or my actual phone/SMS service) initially but I quickly got past it.
Imgur For Android. I take and send a lot of pictures and usually use Imgur from my desktop so the app seemed like a no-brainer. It works reliable and easily, no issues with it at all.
Chrome To Phone. I am an avid Chrome user on my PC so this seemed like a no brainer. Sends links, maps, e-mail addresses, basically anything you're looking at in Chrome to your phone.
One thing I don't use/don't get all the hype and fuss about...
SMS drop-ins. At one point or another since I've had my Optimus S, I've installed GoSMS, ChompSMS and Hancent and I just don't get the hype. If I wanted my text conversations to look like I were having them on an iPhone, I'd buy an iPhone. The standard Android 2.2 messaging app works just fine, in fact, I actually prefer it's predictive text to any of the 3rd party ones I tried.