Grandpops is his whole name and nothing more, so no apostrophe. The cuddle comes from him, Grandpops. If his name was just Grandpop (with no S) then adding an S would mean that there are more than one of him. Heaven forbid, I imagine you thinking.
An apostrophe used to be a way of indicating that something had been shortened by leaving out a few letters. That was way back when the full way of showing possession would have been something rather like "fatheris chair" (shortened to father's chair, which is how we still write it)
Writing "Grandpop's chair" with the apostrophe after the name but before the S would mean the chair belongs to the Grandpop who didn't have an S in his name. For your Grandpops it should really have to be "Grandpops's chair" which sounds daft, so it gets written "Grandpops' chair" with the apostrophe at the end of his name, but the added S left off.
Hope that makes sense.
Or you could just forget apostrophes altogether - they are very small, no-one would miss one.
