You are able to run OBS as administrator on login without elevation prompt by using the task scheduling feature of Windows.
You should find the task scheduler by searching for "scheduled" in Windows Search. It should find your localized name for task scheduler.
In Task Scheduler, right-click the library and select Create new task.
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Set the properties of the task like this:
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It's important to "run only when user is logged on". If you don't do this, OBS will not appear on the desktop but run only in background.
Running with admin privileges is activated with "Run with highest privileges".
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The action in detail:
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The arguments are --startreplaybuffer --minimize-to-tray
The "Start in" directory is important, you have to set it to the directory where obs64.exe is located. If you don't set it, you get errors that several locale files cannot be found.
In the Settings might be of interest. Deactivate "Stop the task if it runs longer than..." if you never terminate OBS.
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This task scheduler trick works for every app you want to start at login. However, it's kind of a security risk.
You can even put an icon on the desktop that, upon click, will start this task manually, so you start OBS Studio as administrator without elevation prompt. Create a new shortcut with this target:
C:\Windows\System32\schtasks.exe /run /tn "OBS Studio autostart"
Notice that "OBS Studio autostart" is the name of the scheduled task you gave on the first property page of the task.