I powered on my Ubuntu computer and see "your computer needs to restart to finish updating."
This indicates that something automatically updated - therefore this is something that wishes my computer to act without first receiving my command.
What could be responsible for this, and how do I kill it?
By Default, theres several automatic Upd/Syn-Functions in the System, to ensure Actuality, Integrity and Security. These are scheduled via "Cron" and/or Event triggered.
The Restart-Request indicates a Change on the Kernel itself or its Components and Drivers.
Software-Center related Updates (Packageindex, installed Programs) can be disabled in the "Software-Updater" Settings. The Firmware/unattended-Ups/Syns for Userspace-Config/Setup by rerouting to "/dev/null" (NOT recommended).
Automatic updates in Ubuntu can be disabled in the "Software & Updates", under the "Updates" tab.