I just purchased a new HP Stream 14 laptop running windows 10 s mode. It will not allow me to download gog galaxy because it is not an "authorized Microsoft app". Is there any other way to download gog galaxy?

Thank you

Congratulations, there's a non-zero chance you bought one of those awful editions of Windows that doesn't let you run external applications.

That or you're getting spooked by the UAC screen.

My recommendation is to shrug and install Linux. Deep dive.

"Windows 10 S" is not really Windows. It's basically the latest version of Windows Phone. It's more akin to Android or iOS than it is to Windows. It's Microsoft's attempt to try to compete in the "Chromebook" market of "web browsers with keyboards". If your hardware is x86 (Intel/AMD CPU), you should be able to find a way to make it real windows, but even then, devices that come with it are usually very limited hardware. If not, you should probably return it as the dud it is.

https://support.microsoft.com/en-us/help/4020089/windows-10-in-s-mode-faq

Galaxy is optional.

I believe there is a way to free upgrade to Windows 10, you have to search how to do it, can't help you further.

Edit: first google result

Edit n2 - rant: We are well pass the point of no return, Microsoft do now whatever can get away with (as ever did) and Windows 10S is another on a long list...

The whole point of Windows 10 S is to lock you to Microsoft Store, ie. you buy any software (including games) only from them. Not from GOG (Galaxy), not from Steam, not from anywhere else besides Microsoft Store. They claim to do this for security reasons (as if getting your games or applications from competing digital stores is riskier than from Microsoft), but in reality Microsoft just wants you to pay THEM for any application or game purchases. They want to prevent competition.

So if you want to play non-Microsoft Store games (like from GOG) on your computer, replace Windows 10 S with something else, like full Windows 10 or even Linux.

And then some still claim Microsoft is not trying to lock Windows users to Microsoft Store, tsk tsk... With small baby steps, nudging Windows users slowly to use only MS Store...

You need regular Windows 10.

I setup a pair of Lenovo Ideapads last summer for my Mother and Brother. They came with Windows 10 S. There was a one way conversion to regular Windows 10. I believe this was done in the Windows store. It was just a matter of clicking a few buttons and waiting. No cost. Very easy and should solve your problem.

It doesn't even matter here, his OS can't run ANY GOG games.

Sorry, just heard that so many times it’s automatic now. Not heard of this s version before, more dreadful M$ decisions. They seriously don’t have anything left they can kill off in the most excruciating ways. And I use a lot of their products. Office, Hotmail, windows, GFWDead, windows store, it’s like a procession of the unholy nowadays.

You know the meme. Hold my beer...

https://support.microsoft.com/en-us/help/4020089/windows-10-in-s-mode-faq

How do I switch out of S mode?
To increase security and performance, Windows 10 in S mode runs only apps from the Microsoft Store. If you want to install an app that isn't available in the Microsoft Store, you'll need to permanently switch out of S mode. There's no charge to switch out of S mode, but you won't be able to turn it back on. If you're blocked from switching and your device belongs to an organization, check with your administrator. Your organization can choose to keep all devices in S mode.

On your PC running Windows 10 in S mode, open Settings > Update & Security > Activation.
Find the Switch to Windows 10 Home or Switch to Windows 10 Pro section, then select the Go to the Store link.

Note
Don't select the link under Upgrade your edition of Windows. That's a different process that will keep you in S mode. Also yes.
Using Galaxy might be beneficial for downloading/installing big games.
Using Galaxy might be necessary for some multiplayer games.