All credit for this discovery should go to Ricci Adams, who told me about it. But Ricci doesn't have a blog, and I do, so here it is. Thank you very much, Ricci, and please start a blog!
Much has already been written about the new monthly screen recording prompt in macOS 15 Sequoia. As always, Michael Tsai has an excellent summary.

The good news is that there's a way to stop the prompts forever. Ricci Adams found the file where the prompt dates are stored.
defaults read ~/Library/Group\ Containers/group.com.apple.replayd/ScreenCaptureApprovals.plist
This file is protected by TCC, so to access it you'll need to grant Full Disk Access to Terminal app.
{
"/Applications/
}
In the plist file, the keys are the paths of the executable files with screen recording permission, and the values are dates. I'm using the Shottr screenshot tool as an example.
To stop the prompts forever—for the rest of your life, anyway—set the date to far in the future, for example, the year 3024 instead of 2024.
defaults write ~/Library/Group\ Containers/group.com.apple.replayd/ScreenCaptureApprovals.plist "/Applications/Shottr.app/Contents/MacOS/Shottr" -date "3024-09-21 12:40:36 +0000"
You'll need to do this for each app, and afterward logout and login again so that the replayd process recognizes the new defaults.
At this point, I typically spoil my blog posts by writing something nasty about Tim Cook, Craig Federighi, and company, so I'll refrain from that this time and just let you enjoy your newfound freedom from monthly nagging.