Having developed many websites for over 8 years, from the days of Mambo and now to Joomla, I have developed a list of preferred must have extensions that enable me to build websites on Joomla easily. This list is my personal preference and I install most of these after a brand new install before any real customisation is done.
1. Editor - JCE
Joomla's default editor TinyMCE while useful is limited in it's current state. That's where JCE comes in. JCE extends your ability to use Joomla with powerful features. From text formatting, image editing to link insertion, JCE is the most powerful editor for Joomla. You can extend it's functionality by installing plugins to upload files, insert media files and more.
There is nothing worse than a hacked, compromised or even a borked website with no working backup. Akeeba is a powerful backup extension that does all the owrk for you. It creates a snashop of your website including the database and an installer that enables you to restore your website easily. Akeeba can backup your site automatically in multiple formats ZIP archives, JPA or JPS archives to clone, move or migrate. The Lazy Backup helps you to automatically backup your site's database and email you every 'x' days and send it your email address or save it on the server. There is detailed documentation for the extension on their website with two versions to choose from Akeeba Free and Akeeba Pro.
Joomla administration can be daunting especially if you are new to Joomla. JSN Admin comes in handy by making managing content in your Joomla site easier. It adds an enhanced Joomla adminstration menu by adding useful links such as uninstall in component menu and a direct install link to the extensions menu. Another useful feature is the spolight search that allows you to search for anything instantly. The third feature is the site manager which allows you to manage your site in one place by managing menu items, components, adding and moving modules all in one place. Try it, you will love it.
4. Admin Tools
There is no such thing as 100% security on the internet and Joomla is prone to hacking as is any CMS but there are extensions that can be used to enhance security of your Joomla site? Admin Tools is one such extensions from the people who brought you Akeeba backup, which allows you check which version of Joomla you are running and update to the latest version. It also allows you to change the dataase prefix, add multile password layers and password protect your Joomla administrator and many more.