Hello fellow partygoer! I have an answer to your question. First, I'll rephrase your question slightly. Instead of estimating the number of sides of your hypothetical die, N, you are really interested in the probability of obtaining a 1, which is 1 / N in your model.
So, we have some experiment that either succeeds or fails, and you know only how many failures have occured in the past. The goal is to give a best guess for the probability of success.
In fact, this has already been solved over 200 years ago (!!) by a guy named Pierre Laplace. He was interested in estimating the probability that the sun will rise tomorrow, given that it has risen for every day for the past, say, n days.
His estimate was that this will occur with probability (n + 1) / (n + 2).
You are interested in the opposite, namely the probability that a success happens, given that only failure has happened for the last n trials. This will then be 1 - (n + 1) / (n + 2) = 1 / (n + 2).
If n = 100, your best guess for the probability that the next trial succeeds is thus 1/102, which is a little bit less than 1%.
for some more info, if you are interested.