I have an iPhone 5s I am repairing,
I want to back it up,
but it has no screen.
Or home button.
I know how to get the pictures off, but can I boot up the iPhone and plug it into my iMac even if it has no screen?
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You should definitely try it. A few additional thoughts:
This may only work if you've previously plugged this iPhone in before and selected to "Trust" this iPhone with the computer already. Otherwise, iTunes is going to tell you that you need to press Trust on the iPhone.
If your mac is running Yosemite or higher, you may find QuickTime handy to mirror the screen display. :)