There are over 7,500 extensions on the Joomla Extensions directory, but these are the ones I recommend and use for your Joomla site as well as my own.
Extensions for Every Joomla Site
These extensions are key ingredients for your website. I install the following extensions in all of my Joomla projects.
Responsive, customizable visual appearance for your website
Firewall and site security
Joomla Content Editor
Full-featured editor used throughout the site
Full website backup/restore
Contact form and other highly customizable forms
Social media sharing in one easy step
Slideshows and image display
Specialized Extensions for Your Joomla Site
The following list includes extensions I have either used for client websites or for my own sites, or both. This list gives you an idea of the many extensions we can add to Joomla to customize your site. Remember, all extensions integrate with Joomla's core functions rather than hacking the core, so the extensions also play well with each other.
Ecommerce cart for both physical and digital products
Newsletter service that works right from your website
Calendar extension with all the features you could want
Optional Registration add-on to JEvents
Blogging extension with Wordpress features that integrate directly into Joomla
Project management extension for individual or group projects
Facebook-like, on-site community software with all the features you'd want
On-site video player
On-site audio player
For the complete list of extensions for Joomla 3.4.x, see http://extensions.joomla.org.