+1 for ES File Explorer. I use it in combination with VPlayer to browse and view videos from my computer.

While it's more a matter of personal preference, I think SwiftKey beats the living crap out of all other mobile keyboards - it's actually the reason my mom decided to get herself an android phone, after seeing me pumping out text like crazy on mine.

Something that's nice to have is a QR scanner - doesn't really matter which one, they all do the same. I just picked the one that had the most appealing UI.

Tapatalk is nice to have if you frequent a bunch of forums *#hint hint @admins*

Popup widget is nice of you want widgets but like to keep a clean home screen.

And let's not forget Tasker. It's a must-have, I don't think I could live without it anymore.