What is the name of your state (only U.S. law)? MN
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Can I use the profile pictures of the top couple thousand YouTubers? I plan to use them for a 'guess the YouTuber'-style app (with ads and a life&coin system). Do I need permission?
Yes, you need permission.
The photographs have a copyright that provides the photographers with all rights to the photos, including reproduction rights. You will need to contact the copyright holders for permission to use the photos.
Additionally, if the rights-holder/photographer is different than the person pictured in the profile (likely), you will need permission from the person pictured in the profile to use their image in a commercial manner.
That is a lot of permissions to get.
Here is a link to Google.com and YouTube's "Frequently Asked Copyright Questions:"
https://support.google.com/youtube/answer/2797449?hl=en