Must be stretched a full link + 1/32 a link . If it were my bike the cassette and chain would have been replaced once per year. That little stretch (is it a regular/daily rider?) over so long time is indeed rare...I would go and replace the casette and chain just as a regular maintenance thing. Not that it will necessarily solve this problem.

Just cus I didn't see it mentioned yet, are there any stiff links in the chain? to me it sounds like cable stretch or friction in the housing, but you seem to have ruled those out. If barrel adjustments don't work and the housing and cables have also never been replaced I would probably just start doing some of the regular maintenance things and see if it stops, if not at least you've eliminated those possibilities.

Then again, I'm not slow to spend money on things..

EDIT: and you are measuring right...should be an inch from pin to pin and no more than 1/8th inch stretch over 12 inches.