This app was mentioned in 4 comments, with an average of 6.25 upvotes
I've been using the same weather widget since Froyo in 2011. I can't believe it still works and is available.
Widgets I use
Aix Weather Widget - Home screen widget that shows forecast of temperature, maximum/minimum precipitation and sun rise/set times graphically compactly and accurate. Free version available here.
Solid Explorer Classic | Solid Explorer Material Design - I use the FTP widget when I need to browse my phone's files. I know I can use AirDroid, but I prefer to use this one.
SoundHound - I have the widget on my lock screen and home screen to swiftly ID songs.
Calculator++ - Can work in a separate window floating over other applications, for when I need to make calculations while remaining in the app I'm currently using.
JOT! - Notes Widget - When I need to quickly write down thoughts, notes, and or ideas.
LMT Launcher - Don't know if this counts, but I use pie controls to quickly access my most used apps (supports up to 24 different apps). Needs root.
Power Toggles - Same as LMT Launcher, but can be placed in the notification bar. I use it to toggle mobile data | wifi | flashlight | airplane mode | brightness | and to enable/disable phone rotation. Supports up to 16 different apps, widgets, actions, etc.
Other widgets mentioned in this thread
Pocket Casts - Listen to podcasts.
Google Search - Search to your curiosity's content.
Sunrise Calendar - Free calendar made for Google Calendar, Exchange and iCloud. Sunrise also connects with all your favorite apps like Facebook, Evernote, Foursquare, TripIt, Todoist, Trello, and more.
DashClock Widget - DashClock is a home screen clock widget for Android 4.2+ phones and tablets, with lock screen support for Android 4.2-4.4. It also exposes additional status items called extensions. The widget comes bundled with extensions that give you instant access to: Your current local weather; Missed calls and unread text messages; Your next calendar appointment; Unread Gmail inbox or priority inbox conversations; Your next scheduled alarm, and more if you install other add-ins/extensions.
Popup Widget 2 - "Very customizable widget for widgets. You can create widget groups to launch from the click of an icon along with a lot of customization." - /u/heavyMGS.
aCalendar Calendar & Tasks | aCalendar free - Feature-rich calendar w/ myriad of options.
Google Keep - Notes & Lists - Capture your thoughts, notes, to-dos, lists, photos, and audio.
/ Added Popup Widget 2, aCalendar, Google Keep
Aix Weather Widget Paid | Free - Basic 24hr weather info displayed as a graph.
JOT! - Notes Widget - Simple notes widget.
SoundHound Music Search Paid | Free - Identifies music playing.
I'm still using Aix Weather Widget after all of these years since it was first posted on here. I love that it displays the weather as a simple graph.
Also a 5x5 Samsung Calendar widget in month-view on my second page.
Power Toggles too. Moreso on my older devices and Android box than on my current phone. Besides using the notification toggles, I really like that I could have the calender day in my notification bar.
SoundHound - Swiftly ID songs. Also like that it can automatically add the songs to a Spotify playlist.
Solid Explorer FTP - To quickly transfer files over Wi-Fi.
And I don't know if it counts, but my device's Always On Display on the lockscreen. RIP lockscreen widgets, gone and still salty about it.