Warning: Blatant spoilers of Red Dwarf follow.
David Lister, star of Red Dwarf, has two of these strange and mysterious things. Apparently, it was "Red Dwarf's most beloved continuity error" (unknown) when it happened.
It all started with Series 2's Thanks for the Memory. When Rimmer had some of Lister's memory, he mentioned having his appendix out twice, indicating that both men had had theirs removed. However, in Series 6, Legion takes Lister's appendix out also - on-screen this time. Hm.
One fan theory was that it was rebuilt when Lister's body was rebuilt in Series 4's episode DNA. However, Doug Naylor, after a falling out with Rob Grant, wrote "Last Human" and pointed out that Lister was born with two appendices. This raises a question after watching Series 7's Ouroboros: since we find out that Lister is his own father (which, even after watching it a million times, still confuses me) then how did Lister get the second appendix? Is this a sign that all is not quite what it seems? Is Lister, in fact, truly some kind of God, as pointed out in Series 1's Waiting For God? It also raises the chicken and egg problem: did baby Lister come first, or adult Lister? What the hell is going on? Is there going to be a ninth series, and if so, will it explain all of this crap? Doug, I need answers!
But back to the appendix. It is indeed mankind's most useless organ. A cul-de-sac of squishy stuff. I once thought that it may have been somehow connected to the umbilical cord, but realised that its position in the body is not close enough to human bellybuttons. Rats.
And the struggle for an answer continues...