Some of the app suggestions here make me cringe.
Here are the apps that I can't live without.
Some of them are preferences that different people may or may not like, but most of these are honestly and objectively the best apps on the market for their purpose. Believe me, I've tried them all. Or don't take my word for it and try them out yourself. Either way, I dare you to show me a better app of its kind.
Solid Explorer Absolutely the best file manager out there. Hands down.
Nova Launcher My absolute favorite launcher. It's incredible.
Helium sync and backup The best app backup and restore tool, for simply backing up and restoring apps.
Droidedit This is pretty much notepad++ for android
Android IRC My favorite IRC client for Android.
Auto sms Great tool that lets you schedule sms messages
Greenify helps battery life by hibernating apps.
Mx Player The best video player for Android.
Pocket Formerly "read it later"
Maxthon Browser Browsers are very much a preference thing, but I think Maxthon is the best out there.
Quickpic The best media browser ever.
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Sideloaded/non play store apps
https://f-droid.org/ = a database of free android software and apps. A wonderful thing to have. This is where you download adaway from.
Black Market This is how you get any paid app you want, for FREE. [I mostly use this to test out paid apps that I want. If I actually find the paid app version useful, I almost always purchase it in the play store, to support the dev.}