The following content relates to the January JoomlaChicago meeting on "Must-Have Extensions." This content parallels the presentation and includes a list of discussed extensions with relevant information, including links to the extensions' sites.
Some extension tips:
Installing extensions is very straightforward and well-documented. Smooth installation depends on a properly set up Web server; these topics are addressed and extensively discussed at the main Joomla Web site (forums et al).
Before installing an extension, back up your site's files and database.
When considering an extension, take the time to evaluate it.
The JED is a great resource: Read the reviews and ratings.
Look at the change logs and how recently the software has been updated
Visit the developer's Web site and check for development and community activity
Nose around and search the discussions at forum.joomla.org
Investigate whether the extension is developed with Joomla's MVC development pattern: If it is, the developer will be much better situated for the move to 1.6, and MVC-based extensions offer other advantages (like support for HTML overrides).
Take the time to read/understand the extension license: most GPL/free extensions are pretty straightforward; commercial extensions may have special or unique terms (including funky stuff, non-U.S. licensing terms, etc.).
Some commercial extensions use DRM (e.g. Ioncube) to “lock” an extension to a domain or server; it's worthwhile to know whether this is present before purchasing. (None of the below featured extensions have DRM, to be best of my knowledge.)
Good commercial extensions are worth the money.
List of Must-Have Extensions
Edit With Panache
Joomla's native WYSIWYG editor is pretty basic: Adding a third-party WYSIWYG editor is a must for any Joomla site: It provides easier ways to handle text, images, file links, etc., as well as improvements to working directly inside the HTML code.
JCE
http://www.joomlacontenteditor.net
Cost: Free, with some free and commercial plugins ($30 annual subscription)
Includes all of the primary features in a WYSIWYG editor, including customizable buttons, group profiles, custom preference settings, image insertion tool, link tool, etc. Plugins can enable some advanced stuff like on-server image and file editing, media insertion, etc.
XHTMLSuite
http://www.xhtmlsuite.com
Cost: ~40 Euros for single site license; other packages available as well
Includes all of the primary WYSIWYG editor features, plus some nice interface enhancements like a context-sensitive Properties area. Also includes lots of controls for group-specific profiles, insertion of specific types of media, etc.
Back Up Your Site
It's not too difficult to manage backups manually via cPanel or rely on your Web host's backup service, but it can be very handy to have your own backup readily accessible, as well as an automated backup process.
JoomlaPack
http://joomlapack.net/
Cost: Free, although the developers are changing versions to a free/paid model soon
JoomlaPack provides more than just a backup of your Joomla-powered Web site: It also builds in a migration process for easily moving a Joomla installation from one location to another. This helps a lot with testing, templating, backups, etc. It supports a variety of configuration and compression options, as well as support for dumping backups to external FTP sites.
XCloner
http://www.joomlaplug.com
Cost: 50 Euros to 100 Euros; note JoomlaChicago discount of 25%
Xcloner is an excellent Joomla backup utility profiled at December's developers meeting. It is extremely feature-rich, including such things as chron job integration for fully automated backups, exclusion filters for files and directories, ability to dump to a remote FTP location (or locally on the server), and many other options.
Manage Files on the Server
Sometimes, it's nice not to have to rely on cPanel or the command line ...
eXtplorer
http://extplorer.sourceforge.net/
Cost: Free
eXtplorer integrates a visual file manager interface inside of Joomla. This allows you to view the files and directory structure of your Web hosting installation, edit/copy/move/delete/create files and directories, creating and extracting archives with files and directories, and changing permissions.
Capture Some Data
Multiple JoomlaChicago members recommended the RSForms Pro extension, so I downloaded it and checked it out.
RSForm!Pro
http://www.rsjoomla.com/joomla-components/joomla-form.html
Cost: ~20 Euros for one domain/support; ~60 Euros for unlimited domains/support
RSForms Pro makes it much easier to manage form creation on your Joomla Web site, as well as tracking the results. It has visual tools available for defining and building form fields, as well as a number of layout templates from which you can work. It can e-mail form results to specified addresses, as well as store form results online for later review or export.
Tweak Those URLs
Joomla's native SEF component does a pretty good job at creating human-readable URLs, but for total control, and for control over META information as well, a more advanced URL manager can be handy.
sh404SEF
http://dev.anything-digital.com/sh404SEF/
Cost: Used to be free; now $35 for one year subscription
sh404SEF has been going through some changes with its recent takeover by Anything Digital, and some users have complained about support issues. Still, though, this component continues to be useful. It allows you to specifically tune the URLs for the pages of your site, as well as the META tags, in order to optimize for search engines. sh404SEF also has a security feature whereby it scans for and removes many common injection-style exploits.
Create Easy Image Pop-Ups
Thumbnails with pop-up images are an often-requested feature on Web sites.
mavik Thumbnails
http://extensions.joomla.org/extensions/photos-a-images/articles-images/6263
Cost: Free
Mavik Thumbnails works as a system-level plugin that makes it extremely easy to create image pop-ups from thumbnails that appear in a nice Joomla modal box. Once the plugin is installed, and published all you have to do is place your pop up-sized image inside of a content editor, then change the pixel dimensions of the image. Mavik Thumbnails will automatically create a clickable pop-up inside a Joomla modal window that appears floating over the page content.
Integrate Other Scripts
Sometimes, the solution to a site requirement is not found inside of Joomla or a Joomla extension. Although Joomla's wrapper menu item can enclose other pages (and Web applications, etc.), this only works for the content/component area of a layout.
Jumi
http://edo.webmaster.am/jumi
Cost: Free
Jumi allows you to integrate outside javascript, php, HTML, css, plain text, etc. files into your Joomla installation directly, from a file, or from a database record. The Jumi code can run in articles, module positions or on its own pages. Uses include integrating snippets all the way to creating new levels of functionality via integrating outside Web applications.

MetaMod
... used to precisely control module appearance according to a number of advanced parameters (including time of day/date, end user's geographical location, presence of specific content on the page, etc.)
Fabrik
... an advanced content creation kit/application builder. I haven't tried this, but it sure looks cool ...
FlexiContent
... also an advanced content creation kit; this appears to be more focused on building new specific types of content. (For example, com_content builds articles, with their standard structure and fields; FlexiContent lets you add and customize the types of fields comprising an article-like item.) I also haven't tried this, but it appears promising for some specific uses.
JoomlaGATor!
... Avery recommended this extension as an essential complement to sh404SEF or any SEO enablement on a site. It looks like it enables linking specific Joomla user activity to Google Analytics, and also helps handle advanced tracking of online marketing campaigns (i.e. who clicked what). I need to play around with this ...
All of these suggested extensions are non-commercial/free GPL, and they can be found at extensions.joomla.org.
Lastly, Joann mentioned an extension to manage updates for other extensions (something like a Joomla-side manager for extension versions). I looked in the JED and couldn't find anything like this. Does anyone know of an extension for this purpose?