Google Opinion Rewards
"Answer quick surveys and earn Google Play credits with Google Opinion Rewards, an app created by Google Consumer Surveys. Download the app and answer basic questions about yourself. We'll then send you surveys around once a week, although it may be more or less frequent. You'll get a notification on your phone when a short and relevant survey is ready for you, and can receive up to $1.00 in Play credits for completing it. Questions can range from, "Which logo is best?" and "Which promotion is most compelling?" to "When do you plan on traveling next?"
Orbot: Proxy with Tor
This one is mainly for privacy concerned users, it may therefore be slower as compared to other VPNs. "Orbot uses Tor to encrypt your Internet traffic and then hides it by bouncing through a series of computers around the world. Tor is free software and an open network that helps you defend against a form of network surveillance that threatens personal freedom and privacy, confidential business activities and relationships, and state security known as traffic analysis."
"Are you having trouble falling asleep? The Twilight app makes your device screen adapt to the time of the day. It filters the flux of blue light emitted by your phone or tablet after sunset and protects your eyes with a soft and pleasant red filter. The filter intensity is smoothly adjusted to the sun cycle based on your local sunset and sunrise times."
LastPass Password Manager
LastPass is a password manager and password generator that locks your passwords and personal information in a secure vault. LastPass autofills web browser and app logins for you, and generates new, secure passwords instantly.
One password is all you need with LastPass – your LastPass master password handles the rest.
WifiMapper - Free Wifi Map
"WifiMapper maps the best free hotspots nearby (from restaurants to libraries to coffee shops and cafes). The details page gives you relevant information on the hotspot and its location in addition to comments from the community. WifiMapper can show you the best hotspots in any major city around the world, great for saving on roaming fees and keeping costs down when you travel."
Podcast & Radio Addict
One of the most downloaded Podcast apps out there. "Podcast Addict allows you to manage Podcasts, Radio on Demand, Audio books, Live stream radios, YouTube channels/playlists and RSS News feeds from a single app."
ROM Toolbox lite (requires ROOT)
"ROM Toolbox combines apps like Titanium Backup, Root Explorer, SetCPU, MetaMorph, Autorun Manager, Terminal Emulator, Scripter, SD Boost, BuildProp Editor, Font Installer, Boot Animations & many more apps into an all-in-one app!" It is capable of performing most of the tasks that require your device to be rooted.
Betternet Free VPN Proxy
This one is good to use for bypassing bans and surprisingly, unlike other VPN apps/services it doesn't require registration or doesn't have a usage limit.
VLC for Android
"VLC media player is a free and open source cross-platform multimedia player that plays most multimedia files as well as discs, devices, and network streaming protocols. It can play any video and audio files, as well as network streams, network shares and drives, and DVD ISOs, like the desktop version of VLC."
Opera Max - Data saving app
This app is for saving and monitoring data being used by applications installed on your device. It is simple to use and it doesn't affect encrypted traffic. It establishes a VPN connection and compresses the data which is then sent to your phone saving you a lot of data usage. Great app if limited data usage is an issue for you.
FP edit: Just corrected a few typos and errors. Thanks all for the upvotes! Hopefully, you'll find them useful :)