I am a heavy user of Google Apps including Chrome and Chromebooks. These are the extensions that I use and feel are some of the best and most useful and consider a must have for all Chromebook users. Title of extension links to Google Play store. Link at end of entry is to other articles on how to use it.
Clearly - Clearly makes blog posts, articles and webpages clean and easy to read. Save them to Evernote to read anywhere. - great for students as it eliminates distractions on the web pages and makes it easier to read and understand. Evernote Clearly
Evernote Web Clipper - Use the Evernote extension to save things you see on the web into your Evernote account. - save web sites, online files and more. A must have. Evernote for Education Resources.
goo.gl, the Google URL shortener - shorten URLs, and share with many different services! - Very useful for sharing links with others, especially if you want them to write it down. Google URL Shortener Features
Google Docs, Slides, Sheets - create, view, edit and share documents, presentations and spreadsheets with others. Another must have. Google Apps Resources for Education
Google Keep - notes and lists - Quickly capture what's on your mind and share those thoughts with friends and family. - great for quick notes and reminders. Google Keep Features
Google Translate - View translations easily as you browse the web. Excellent for ELL students and Foreign Language classes. Google Translate features and languages.
Office Editing for Docs, Sheets & Slides - View and edit Microsoft Word, Excel, and PowerPoint files with Google Docs, Sheets, and Slides - another must have. Who needs Office anymore?! Google Docs Editing for Office
Pocket - save articles, videos and more to read later. I save things from my Feedly account here to read later and then can save from here directly into Evernote. Pocket is one of my most used Apps.
Save to Google Drive - Save web content or screen capture directly to Google Drive. Great for saving online files right into your drive account.
Screencastify - simple video screen capture software for Chrome. Use to record lessons and classes for students to access later.
Webpage Screenshot - The most powerful and freeware app for capture entire page, Save & Edit & Annotate & Print & Share - I use this to capture screenshots for blog posts.
WeVideo Next - video creation and editing tools using HTML5. Great tool for using videos on a Chromebook. WeVideo Online Video Editor
WeVideo Next - video creation and editing tools using HTML5. Great tool for using videos on a Chromebook. WeVideo Online Video Editor
Related:
Resources to get Started With Google Apps and Chromebooks
Google Apps and Chromebooks Web App Recommendations - a whole lot!
Google App and Chromebooks Resources - learn how to better use them