I like Cobian Amanita for backups. It has all the features you'd want. The most impressive feature for me is the ability to use Volume Shadow Copy which makes it possible to backup open files. I run it on my file server at work to backup the network folders to an eSATA drive every night.
http://www.cobiansoft.com/
Off the top of my head, a quick list of the other free programs I like:
IMGBurn (Disc Burning SW)
Microsoft Security Essentials (AV)
Avast Home (AV) < I used to think it was the best but got two Trojans while running it so now I run MSE.
OpenOffice.org (Sun's free MS Office alternative)
Malware-Bytes (Malware Removal Tool)
Firefox (browser)
Utorrent (Bit Torrent Client)
Imagresize.exe (Powertoy from MS for XP to enable easy right-click batch resize of images)
VSO Image Resizer (Similar to Imageresize.exe for newer OS's)
PDFCreator (Print to PDF, etc.)
Foxit Reader (Alternative to Adobe Reader) <Watch out, installs the Ask browser tool bar.
Clonezilla (Alternative to Ghost)
Seagate DiscWizard (Rebranded Acronis disc imaging SW)
gparted (Disc Partitioning boot disc)
Handbrake (Movie Ripping SW) <Never actually used it.
MPC-HC (Media Player that supports hardware acceleration)
VLC Player (Media Player that plays everything but no HWA)
Haali Media Splitter (Supposedly all you need to play every video with WMP or WMC in Windows 7) <So far, seems to be true
Tversity (Streaming audio/video to your PS3 or other media extender, HTPC, etc.) <Didn't have the best quality on my "G" network to my PS3 a year ago, haven't revisited it yet.