There's no need for a single person to digitally sign a PDF file multiple times. A signature applies to the entire file as a whole, not just to the page it appears on.
I agree, but our company wants each page on the pdf digitally signed and it is annoying to sign page 1, save, overwrite, sign page 2, save, overwrite, etc. Right now after I sign the first page a javascrite runs that makes the other pages look like they are signed, but it's basically just unhidding a jpeg image of the signature....the problem is it only works if I sign it. If someone else has to sign it still shows only my signature on pages 2 and 3. The people who put this in place don't understand how acrobat signatures work, kinda stuck in the 90's here.
Then you should try to educate them... To your point, yes, a script can sign (and even save) a Digital Signature field, but it's not simple. Read up on the signatureSign method of the Field object to see how it's done.
This is not just annoying, it is completely and fundamentally wrong. And for any more than a few pages it is a potential catastrophe.