I really love this, and I'm not sure I can get all the reasons why out in a coherent manner.
First, your subject matter. I've been reading a lot about disappearances lately, and you can't get much more high-profile than the leader of a country. Always wondered what it must have been like to be Australian at that time. I mean, Americans have lost Presidents before, but we've always known why: someone shot 'em. To just have him vanish into the tide, that must have been terrifying.
Second, I would totally do this. >.> No, really, living forever sounds great, I fear death.
Third, I will say that I started getting a little apprehensive at the list of instances. I mean, in some ways, it's a laundry list of GoIs and other skips applied to this one guy. It does make the point that he's somehow attained immortality in every way imaginable, and I lost the apprehension by the end, but it's worth pointing out.
(Also, I've had an idea about people lighting themselves on fire and burning for an extended period of time for a while now. Maybe we can talk later.)
Lastly, I really like the "they all react the same". They aren't a hive-mind, that much is clear, and the fact that 13 and 14 greet each other differently in that big round-table meeting is what makes the log work. I feel like "what if they didn't turn out to be different people despite their different experiences" is somehow part of the anomaly as well.
So yeah, long ramble, a few typos, but neat article! +1