The normal distribution can be derived from basic principles and calculus The Normal Distribution: A derivation from basic principles. Are there other distributions that can be derived like this from naturally occurring processes like those found in simple games?
Like any sort of familiar games specifically such as backgammon, billiards, cards, checkers, or chess to name a few.
This is the case of the majority of the most common distributions. If you want to know more about this, I would recommend that you watch the youtube videos on special distributions from Stat 110 at Harvard. The teacher does a great job at linking the distribution with idealized "naturally occurring processes".
As an example, you might be interested in the characterization of the Poisson distribution as the count of the number of arrivals in some time interval of a Poisson process.
The wikipedia article on the Poisson distribution provides a list of what you would call sequences of events which can be thought of as approximative Poisson processes. In your words, your could "derive" the poisson distribution from these actual sequences of events just as much as you could "derive" the normal distribution from an actual version of the dart experiment described in your reference.