How to estimate the number of articles on Wikipedia using the "random article" function?

If $S$ is the total number of articles, the probability of observing $N$ is $$\frac{(S-1)^{N-1}}{S^N}.$$

We should choose an estimate for $S$ that maximizes this probability. Differentiating gives $$\frac{(N-1)(S-1)^{N-2} S^N - N S^{N-1} (S-1)^{N-1}}{S^{2N}} = (S-1)^{N-2} \frac{(N-1)S - N (S-1)}{S^{N+1}}$$

which is equal to $0$ when $S = 1$ and when $S = N$; the former is a local minimum and the latter is a local maximum, so indeed $S = N$ is a sensible estimate.

However, if you were actually going to try to do this, I think waiting until the first article repeats is a bad idea. You have to wait very long and in the meantime you're not using information about any other articles that may have repeated. As Ross says in the comments another idea is to wait until some article repeats. If $S$ is the total number of articles, the probability of observing $N$ is now $$P(S) = \frac{(S-1)(S-2)\cdots(S-N+1)N}{S^N} = \frac{N(S-1)!}{S^N (S-N)!}.$$

Stirling's approximation gives $$P(S) \approx C_N \frac{ (S-1)^{S-1} }{S^N (S-N)^{S-N} }$$

where $C_N$ is a constant depending only on $N$. Taking logarithms gives $$\log P(S) \approx \log C_N + (S-1) \log (S-1) - N \log S - (S-N) \log (S-N)$$

and taking derivatives gives $$\frac{P'(S)}{P(S)} \approx 1 + \log (S-1) - \frac{N}{S} - 1 - \log (S-N).$$

A local maximum should therefore occur when $\log (S-1) - \log (S-N) \cong \frac{N}{S}$, or when $$1 + \frac{N-1}{S-N} \approx e^{ \frac{N}{S} } \approx 1 + \frac{N}{S}$$

giving $S(N-1) \approx N(S-N)$ or $S \approx N^2$. So as Ross indicates this is much more efficient for obtaining an estimate.