I need help with this following sentence: Mr. K's dogs' babies' grandchildren had birthdays.
Is this where the apostrophes go?
What a very peculiar sentence!
But yes, the apostrophes are in the right places.
It is not necessarily right, unfortunately. The sentence presupposes that there is more than one dog belonging to Mr K that has had "babies". We won't go into the fact dogs have puppies, not babies, however, if there was only one "matriarchal" dog then the apostrophe should be between the g and the s ie dog's. This is the way an apostrophe signifies singular and plural.
Hope that helps a little.