Chrome for Android. This counts for me even though it came with my Nexus 7 because it still doesn't come with most Android devices, even ICS and JB ones, and it's the best browser for Android IMO.
VLC. Plays ******* anything. Can't go wrong.
Amazon Kindle. Best ereader app out there. Not perfect, but still very good overall.
Note Everything. I've not tested out many note taking apps but this one is very good for me so far, I use it in classes a lot.
Google Drive/Google Docs. I do wish it let me create new documents offline but other than that it's brilliant for working on the go.
BitTorrent. Self-explanatory. Awesome torrent client.
FetLife for Android. It's, well, an app for checking FetLife. On Android. Also quite self-explanatory. It's also great to have this on Android since Apple would never allow it on iOS.
ASTRO File Browser. It's become a bit bloated recently and I know a lot of people have moved to ES File Browser but I still love my ASTRO.
AdAway. Blocks ads systemwide. First thing I install after rooting.
DroidWall. Very useful system utility. Allows you to choose which apps can and can't access the internet. If you have data limits it's even more useful, especially since it lets you set apps so they can access the internet via WiFi but not 3G/4G.