OK so let's say that I want to run a relatively stable linux install (so not the arch bleeding edge methodology), but I do want to run the latest linux kernel, or as close to it as possible.
What are my choices? Or is the only choice to alter my interpretation of the word "stable"? :-)
Ubuntu should have a PPA for that
Yup:
https://wiki.ubuntu.com/Kernel/MainlineBuilds
https://kernel.ubuntu.com/~kernel-ppa/mainline/?C=M;O=D
Technically, it's not a "PPA" and isn't usable as a source, but you can download builds and install them.
https://github.com/pimlie/ubuntu-mainline-kernel.sh is a decent script that wraps managing updates and installs.