Minimal distros are a great way to understand what a distribution is made of, and how it interfaces with the kernel.
"Serious" minimal distros usually target embedded systems like phones, cars, routers, etc. in which storage and computing capacity is limited.
https://github.com/ivandavidov/minimal
http://stackoverflow.com/a/30056630/895245
By default, this distribution only boots the RAM initrd: anything you do is lost after shutdown.
System to build distro images locally
Similar to Buildroot.
Supported by the Linux Foundation.
https://en.wikipedia.org/wiki/Alpine_Linux
One step beyond Minimal Linux Live: includes a libc and a package manager with many, many packages.
http://www.linuxfromscratch.org/
Linux from scratch, mostly educational.
Teaches how to build a minimal Linux distro from base standard packages.
Not very automated, although there is an automated version at: http://www.linuxfromscratch.org/alfs/
https://en.wikipedia.org/wiki/Damn_Small_Linux
Seemed popular, but died in 2008?
Tiny core Linux: http://tinycorelinux.net/, source: http://git.tinycorelinux.net/index.cgi Minimalistic with X.
Minimalistic, text based, security: http://rlsd2.dimakrasner.com/
ARM Embedded Linux.
Linaro ARM board bring-up helper.
http://www.linaro.org/blog/community-blog/is-linaro-a-distribution/