Quickie: What update do I need to be able to install patch a Server 2012 r2 RTM machine?

You need...

The current servicing stack update mentioned here:
ADV990001 - Security Update Guide - Microsoft - Latest Servicing Stack Updates

For September 2023, the SSU is:
KB5030329: Servicing stack update for Windows Server 2012 R2: September 12, 2023 - Microsoft Support

Then this KB to be able to install the current monthly rollup:
KB2919355: April 2014 - Microsoft Support

Note: This is NOT every update the machine will need.  It gets you 90% of the way there with basic functionality so Windows Update doesn't crash and can successfully pull in the additional 108 outstanding required and optional updates. :)