It might be fun to make a source-based distribution where every binary in /usr/bin started off as a link to a script that built and installed the requested executable (over the top of the link), before executing it.
Various research OSs are JIT-based, of course. It looks like JX (a Java operating system) caches its native code, so it's not 'pure JIT' https://github.com/mczero80/jx/blob/5fbeae79/libs/compiler_e...
It looks like Cosmos (a C# operating system) does the same https://en.wikipedia.org/wiki/IL2CPU
I think it would need to be integrated into the package management system pretty tightly (or have one of its own) to get all of the shared library dependencies.