Double Eagle's Bye Hole comes after the ninth green, on the way to the clubhouse and the 10th tee. I think it's reversible, meaning players also have the option of playing it as they come off the 18th green and walk around the clubhouse, but it's far easier to sequence in between nines.
Devil's Pulpit north of Toronto has three 11th holes, one of which plays as a Bye Hole to get to the 12th tee after you've played one of the 11ths. Don't ask. When you're that rich, you can do anything you want. Sounds like fun to me.
Anyone who tells me it's not within the rules to post a score when playing like that has a lot worse to account for, like the 90% of golfers who don't follow the rules of golf as a matter of course when they play and nonetheless post.