I think mine would be a tossup between WolframAlpha, for just being a bro for all my bath needs, and ventrillo, which while expensive as an app makes a lonely trip more fun when I can talk to a bunch of friends at once.
I have fully switched from Mail/Safari/Apple maps to the Google alternatives. They work brilliantly, especially together. Everynote, Feedly, and Google hangouts are the top apps that I use. I don't have any super special life-changing apps.
I'm on iOS, so these are going to be iOS only: Alien Blue: I use this on almost a daily basis. It's pretty much the best Reddit app for any platform. Now I need a Hubski version. iStudiez Pro: A great app for keeping track of classes, homework, quizzes, etc. There's also an OSX version that syncs up with it. Pandora: I have a modified version that removes ads and gives unlimited skips. WolframAlpha: I snagged it while it was still free. I don't know how the new version is though. Chrome: Pretty much replaced safari for me. Though I use safari on OSX. Edit: Also, Game Dev Story. It's available for both Android and iOS. It's a fun game. Kept me hooked for days.