http://mathoverflow.net/questions/14964/estimate-population-size-based-on-repeated-observation asks the following question.
I take the bus to work every day. Every bus has a serial number, but unlike in the German Tank Problem, I don't know if they are numbered uniformly $1...n$.
Suppose the first $k$ buses are all different, but on day $k+1$ I take one I've been on before. What is the best estimate for the total number of buses?
The provided answer gives a maximum likelihood estimator as well as an unbiased estimator of $k(k+1)/2$
If you know the number of buses can't be larger than some given value $N \geq k+1$, how does that change the maximum likelihood estimator and the unbiased estimator ?