Vtyjkyuk

AFAIK, Poisson Process is an object that takes in an event rate $R$, and produces a time-series of events, such that the expected number of events within any time interval $Delta t$ is $R Delta t$. The rate $R(t)$ may also depend on time, and can be estimated by averaging events within a time window if the changes are not too rapid.

Now, Poisson process assumes that there is no structure in the events, other than their rate. Assume I have noisy random event data from an experiment, and I would like to explore the data for potential additional structure. What is the next simplest process to try, that has, say, two input parameters?

If I turn on my imagination, I am thinking similarities with Taylor series. In a truncated series expansion, every next term increases the accuracy, but also requires better knowledge of the function (experimentally - more data) to be estimated. Is there something similar to series expansion for random processes? For example, in terms of moments - mean, variance, skew, etc.

I'd say that you are looking for a Poisson Process with branches.

Consider for example a Poisson Process $N(t)_t$ having rate $lambda$, and suppose that each event is classified into either one of $n$ types. If you assume that the classification of an event is independent to the classification of any other event, an event is classified as a type $j$ with probability $p_j$ and $N_j(t)_t$ is the process that counts the number of type $j$ events, then
$$N(t)=sum_j=1^nN_j(t).$$
Moreover, it can be proved that $N_j(t)_t$ is also a Poisson Process whose rate is $lambda p_j$ and each $N_j(t)_t$ is independent of every $N_k(t)_t$ with $kneq j$.

If you need not too rigurous book to read stuff like this you can check "Introduction to Probability Models" by Sheldon M. Ross.