See the wikipedia article on the German tank problem for a good story leading up to why you care about them, or for an even simpler one:

Suppose there's a bucket full of N tickets, labelled 1 through N. You don't know N, but you pull out a ticket at random, and see that it has n written on it. What do you suppose N is?

The maximum likelihood estimate says n, which intuitively, to me, is utter stupidity. The Bayesian story sort of confuses me, too. Suppose I had a prior belief that N is uniformly randomly chosen between 1 and 100, and suppose I draw n=2. Then I can rule out N=1, and I believe then that

Pr(N=3|n=2) = Pr(n=2|N=3)Pr(N=3)/Pr(n=3) = (1/3)(1/100)(1/(1/100 + 1/200 + 1/300)) = 1/(3 + 3/2 + 1) = 2/11

etc.

?

[This calculation is utterly wrong, thanks to Susan for pointing it out, still the conclusion I drew from it by coincidence is still right, which is:]

In that case the most probable size of N is allegedly still 2, which seems wrong.

But the MVUE says we should estimate N = 2n-1. Clearly being inutitively right is not the same thing as being formally right, but I'm compelled to consider MVUE an

*interesting*idea at least because it delivers intuitively correct answers when other formal things I already know about fail to.