Jesse Vincent who is responsible for a number of Kindle hacks in the past (such as Savory and tethering Kindle 2) managed to get Ubuntu Linux running on the Kindle 2. Since one branch of the popular distro is ported to run on the ARM architecture that Kindle is based on, it seems like it was should have been possible to run a generic Linux version on the device. Jesse proved it to be true. He has actually been at it for quite a while, having run xdaliclock on the Kindle months ago. It seems that recently he was able make most of the Kindle hardware (like 5-way controller) work.
Hypothetically, there is no limit to what other software could be installed. It’s even plausible that at some point in the future something like this could become more widespread. If someone likes the Kindle but not Amazon’s platform, they could install some community supported Kindle OS that was more to their liking. People already jailbreak iPhones; this could be the Kindle equivalent.