If some ill were to befall my Mac mini, these are the first applications I would put back on after Adobe CS2 and Microsoft Office. Feel free to share yours.
Adium
When I worked on the Windows platform, I purchased a copy of Trillian, an application that linked several different Instant Messenger clients together in one shared contact list. Sadly, there wasn't a Mac version available when I made the switch, so Adium has filled the void. I am not online nearly as much as I had been while living on campus, but it still serves my connectivity needs.
Butler
My Dock still has 24 icons in it, but this little application keeps it from being overburdened. A quick Command-Space and two or three letters allows me to start any application, setting or a host of other items available through its intuitive interface. It even "learns" which applications you prefer launching. I haven't played around with some of its more advanced features, but I am yet to find something it cannot handle.
Cyberduck
Cyberduck handles all of my FTP/SFTP transfers to this and other Web sites. If you can look past the bathtub duck, it really is an easy to use application. My favorite feature of Cyberduck is that it integrates with Smultron (text editor) to do a faux "live edit" on remote text files. This proves particularly useful for Web developers.
Flip4Mac
When Microsoft announced that Media Player 9 would be the last update for the Mac OS, Flip4Mac's plug-in became the preferred solution to playing WMV files within a browser window. It works as a go-between for Media Player and Apple's own Quicktime. Not every embedded player on the Internet works with it yet, but it is improving.
Firefox
I liked Firefox long before my Windows box gave way to the mini. Microsoft's Internet Explorer was showing its age, and every nightly newscast seemed to have a doom-and-gloom story about the latest Internet worm causing havoc for IE users. Firefox too has had vulnerabilities, but it is a far more modern and secure browser. I prefer it over the discontinued IE for Mac or Apple's own Safari.
Gmail Notifier
I am one of those power-users that sets their mail application to check for new messages every 5 minutes. Gmail notifier goes a step further and polls the server almost in real-time. I started using Gmail roughly two years ago, and it is still in my mind the best of the so-called "Web 2.0" applications.
Google Earth
Another one from the Google collection. I cannot think of a single reason anyone would buy a mapping application off of the store shelf. Aside from being a powerful direction-finding tool, Google Earth is fun to explore. With the open-ended development model, a user can import several data layers, in effect creating anything from a directory of diners across America to creating a true-to-scale skyline of their hometown.
Paparazzi!
This is one of those niche applications that I do not use every day, but comes in handy when needed. Until I found Paparazzi!, anytime I needed a screenshot of a Web site, I was limited to the viewable area of the screen. This application allows users to capture the entire page (based on your constraints) and export the image in a wide variety of formats. It is great to have in planning redesign projects.
Smultron
Last but not least, Smultron. This handy text-editor integrates tightly with Cyberduck, so there is not a lot of needless switching and dialog boxes. The left menu bar keeps track of open files (as opposed to horizontal tabs), so it is much easier to switch between multiple files. It also conserves valuable clock cycles by automatically closing when you close the application window. It is light weight enough that opening a document takes only a few seconds even if the application is not running at the time.
Final thoughts ...
Looking for more open-source or free software? Check out OpenSourceMac.org.