So, every time I start my PC this window pops up:
This even happens when Steam downloaded no update. What can I do to stop it from reappearing?
Steam will always check for updates on startup and download and install them automatically to ensure that you're running the latest version. Additionally, Steam will occasionally verify all of its files, to ensure that everything is as it should be. This is the window that you're getting.
This window is displayed while the update is installed or verification process is running to provide feedback to you (the user) that something is actually happening, rather than staying completely silent, leading you to think that the application hasn't started.
If the window appears every single time you start up Steam, you're probably experiencing an issue with Steam realising it has updated.
It's worth noting that this is a well discussed issue with no official solutions available, there are plenty of different suggestions on how this can be fixed though and some of these work for some of the people experiencing the problem some of the time, so you'll want to try all of them if you really want to stop Steam from verifying every time you start it.
Anti Virus/Internet Security applications
This issue can be exacerbated by anti-virus/internet security applications blocking Steam from doing what it wants to and reading all of Steam's files as the Steam application attempts to verify them. Try adding exceptions in these applications for your Steam folder and restart Steam to see if this resolves your issue.
Opt out of Steam beta
If you are a member of the Steam beta program, it might be worthwhile opting out of the beta program and reverting to a previous version of Steam (it should do this automatically) to see if this resolves your problem.
Steam not correctly closing when you shut down your PC
Another thing that is mentioned in the Steam forums, is allowing Steam to verify all of its files and then closing Steam properly (rather than letting Windows kill it as it shuts down). What you'd do here is wait however long it normally takes to verify files and get into the main Steam interface, and then close Steam with the menu option (or system tray icon) before restarting it. After restarting in this manner, the file verification window shouldn't appear, since Steam would have properly closed all of its files when it exited.
Performing a partial reinstall (keeping your games)
If you're certain this is not an issue with your anti-virus/internet security application and you haven't opted in to the beta, you can try one of the following to see if it resolves your problem:
This will essentially reinstall Steam without requiring you to redownload all of your games or losing any of your locally stored settings.
What to do if none of this works
If you're still having problems after trying all of the above, there are really only two options;
With Steam open, click the Steam menu, and choose Settings.
"Run Steam when my computer starts."
I ran into this pattern when I realized that all my computer boots were only because of an unplanned, interrupted reboot. In other words, my computer BSOD'd while Steam was sitting there running in the background in the System Notification Area. This validation, I concluded, was just a sanity check to be sure that the unexpected killing of Steam didn't corrupt any of its files.