A Ubuntu-derived distribution, Puredyne bills itself as an "operating system for creative multimedia" designed to run on a USB drive.
More specifically, Puredyne contains free software tools for audio, graphics and multimedia, as well as for streaming and version control. In each of these categories, the tools range from command line tools to the latest available on the desktop. Its graphic tools, for example, include both ImageMagick for batch processing files from a terminal, GIMP for photo editing and Inkscape for creating your own graphics.
Just as importantly, Puredyne includes a range of equivalent tools. Version control tools, for instance, include Subversion, Mercurial, and Bzr, only one of which any user is likely to want. This variety makes Puredyne a showcase of free software tools, as well as an environment well suited for productivity.
No list of Live Disks would be complete without a forensics disk for investigating system failures and restoring them. CAINE (Computer Aided INvestigative Enivronment) is only a few months old so far as I can tell, but worth mentioning for the thoroughness of its selection of applications.
The function of this Live Disk is self-explanatory. What is less obvious until you try it is Redo's versatility. It can be used on both Linux and Windows, stand-alone machines and networks, and not only for backup, but also for recovering lost files and configuring partitions. Among these tools, the backup is especially noteworthy for its simplicity. Needed drivers or documentation can be downloaded as well via Redo's own Internet connection.
Sugar began as the interface for the One Laptop Per Child project. However, for several years, it has been an independent project overseen by Sugar Labs, which develops a desktop environment for early childhood education. Sugar on a Stick is a Fedora re-spin whose goal is to encourage the use of Sugar among its target audience.
From the perspective of a user familiar with standard desktops like GNOME or KDE, Sugar is a radical simplification. Its home view displays available activities -- the term used rather than application to emphasize the educational aspects -- in a circle. Activities are launched full-screen. A unique feature is its neighborhood view, which shows nearby Sugar users with whom users could interact.
Those with children might want to investigate Sugar on a Stick as a way to introduce them to computers.
Do you want to explore security and privacy? If so, then Tails is one of the most painless places to start.
Tails is a live system designed for safe and anonymous Internet use via the Tor Project. The makers of Tails, many of whom prefer to remain anonymous, have made security and privacy available to anyone who can boot a Live Disk. It even includes a camouflage option when you are using it in public, disguising itself as Windows XP to deflect curiosity from passersby.
However, as useful as Tails is in itself, what may be even more valuable are the online help provided by the project and the links to Tor's documentation. Half an hour's reading will leave you aware of the legitimate reasons for privacy and the basic principles needed -- all of which you can then see in action via Tail's Live CD.
These are only a sampling of the Linux Live Disks available, but they give some sense of the variety.
Almost certainly, the list is incomplete, but it gives a sense of how much Live Disks are a part of free software culture. Somewhere on that list, you are sure to find one that interests you. Live Disks are so much a part of the user culture that, sooner or later, you will find yourself using one -- and probably several.