Any other Linux distribution that has a live image.
Linux
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Linux is a family of open source Unix-like operating systems based on the Linux kernel, an operating system kernel first released on September 17, 1991 by Linus Torvalds. Linux is typically packaged in a Linux distribution (or distro for short).
Distributions include the Linux kernel and supporting system software and libraries, many of which are provided by the GNU Project. Many Linux distributions use the word "Linux" in their name, but the Free Software Foundation uses the name GNU/Linux to emphasize the importance of GNU software, causing some controversy.
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They were specific in their post.
Puppy and Porteux
I've only tried Puppy but PorteuX offers more DEs.
Both of these offer persistence. With Puppy you can choose when you shutdown to save. I think PorteuX works the same way.
TinyCore does this, I think; by default files and applications go into session storage (cleared on logout), but they can be moved/writted to persistent storage. I have to say I digged it, and I wish the driver and application support was better (but then it wouldn't be so minimal)