From a flavor standpoint, you could add just about anything in. Mashed potato is good vessel for a lot of flavors. Personally, I'm a huge fan of brandade (cod and olive oil added usually) scooped on a nice crostini. But for a more typical mashed potato side: butter, milk (or sometimes sour cream - makes it thicker but good flavor), and S&P are enough. Maybe some grated parm or a soft cheese.

The egg is odd to me because I think of it as a binding agent. I would mix a beaten egg with leftover mashed potatoes and cheese to make mashed potato cakes or into a ball, covered in bread crumbs, and fried. The egg holds it together. Not sure I want that in regular mashed, but based on this thread maybe it works out.

And I 100% agree with the poster who mentioned it's just a good reason to eat gravy!