Among the 10,000 (and even more) extensions that are listed in the JED, here are my 10 favorite ones. This is only my opinion, so do not hesitate to discover and to test (on a test site first) the various other extensions that are listed in the JED.
Akeeba Backup is an extension enabling you to back up your website with only a few clicks, and then to restore it in case a problem arises.
For more information on how this extension works, you can read the Backup, Move, Restore a website chapter.
You can download Akeeba Backup from AkeebaBackup.com.
JCE Editor (Joomla Content Editor) is an editor based on TinyMCE, Joomla!'s default editor. JCE Editor is much more complete and offers, among other features:
An improved image manager interface. It is for example possible to add new images to the Joomla! image manager when editing an article (or any other item using a text editor) simply by dragging and dropping.
An advanced user management system. You can for example create user groups (independent of the Joomla! user groups) and customize the editor (and consequently the features) for these specific groups. JCE can also automatically create an image directory so that users access their own images only.
More features such as the popup window on image click.
You can download JCE Editor from the author's website.
Community Builder is an extension enabling you to manage your site's users in a much more flexible way than what Joomla! offers by default. Thanks to this type of extension, you can build a real social network. It takes quite a lot of time to get used to this extension.
You can download Community Builder from Joomlapolis.com.
Komento is an extension managing article comments. This extension is compatible with many other extensions such as K2, Community Builder, Zoo, Virtuemart...
Komento is the extension that manages the article comments on this site, it can be downloaded from Stackideas.com.
AcyMailing is one of the most complete extension for Newsletters. This component allows you to manage subscribers by groups, to send Newsletters to unregistered people, to send joint files... Acymailing is also compatible with Community Builder.
You can download Acymailing from Acyba.com
CK extensions allow you to add new features to your website, like a drop-down menu (thanks to MaxiMenuCK), a slideshow (thanks to SlideshowCK) and you can also create your own template with Template CreatorCK.
All these extensions can be downloaded from joomlack.fr.
Although it takes some time to get used to BreezingForms, this extension allows you to do nearly any kind of form. The latest version of this extension is commercial but, every time there's an upgrade, you can get the second to last version for free. If you find this extension a little bit too complex, you can have a look at these JED categories: Contact Forms, Forms.
If you bought this book in its ebook format, you actually did it via the BreezingForms extension. You can download BreezingForms from Crosstec.de.
K2 is a powerful CCK (Content Construction Kit) broadening Joomla!'s capacity (e.g. in the category and article managers). Although it's not perfect (it doesn't fully support Joomla! ACL system), K2 is very easy to use and you can get great results quickly.
You can download K2 from getk2.org.
Modules Anywhere allows you to broaden the modules publishing options (usually modules can only be published on your template's module positions) and to place your modules anywhere you can enter text.
Please note that the vast majority of the extensions offered by Nonumber are very interesting.
You can download ModulesAnywhere from nonumber.nl.
Phoca extensions allow you to perform many different actions such as managing an image gallery with Phoca Gallery, managing your downloads with Phoca Download, manage your documentations with Phoca Documentation, and much more.
All these extensions can be downloaded from phoca.cz.
Feel free to share your favorite Joomla! extensions in the article comment boxes.