It was a 100th-birthday party, not (barring coincidence) a 100th birthday party, at which Trent Lott said those nice things about Strom Thurmond that cost Lott his Senate leadership position a couple of years ago.
Am I being too fussy in insisting on that hyphen? No.
Everybody knows that the reference is to 100 birthdays, not 100 parties, but is that a principle that can be applied to all such instances? Of course not. Only a 1-year-old can have a first-birthday party, but you may have had your first birthday party at 2 or 3 or 47. The icing-on-the-face thing gets pretty old somewhere between 3 and 47.