If you use Chrome, "Chrome to Phone" is an invaluable service to get stuff on your phone from your computer.
With that, Google Drive is a good way to share content between all your device, or be able to access your data from anywhere (queue rant about privacy, blah blah blah.)
MightyText which allows you to use texting on your computer instead of your phone (when you're using it).
WeatherChannel app gives a decent preview of what the weather might be (even in the notification area, so you can just look at it). Note, like any major weather corporation, the data is usually off. This one is at least close to accurate.
Circle of 6 is a good app for those who are either paranoid or think they may be in danger, as it is a good way to stay safe. With a push of a button, it will notify up to 6 people that you need help immediately.
FoxFi mobile tethering to allow you to tether without your ISP noticing that you're tethering.
WebMD for Android or iPhone: So you can find out that you have cancer from any symptom, on the go!
Spotify for Music, especially if you have T-mobile.
Google Translate is pretty invaluable when travelling abroad.
Technomadia's apps are pretty cool when travelling around (paid):
http://www.technomadia.com/2010/12/introducing-coverage-for-iphone-ipad/
http://www.technomadia.com/2014/04/introducing-us-public-lands-for-iphone-ipad/
Now if you want Chrome apps and extensions... I got a handful of them too. 8)