Examples of processes that are not Poisson? I am looking for some good examples of situations that are ill-suited to model with a Poisson distribution, to help me explain the Poisson distribution to students. 
One commonly uses the number of customers arriving at a store in a time interval as an example that can be modeled by a Poisson distribution. I'm looking for a counterexample in a similar vein, i.e., a situation that can be regarded as a positive count process in continuous time which is clearly not Poisson. 
The situation should ideally be as simple and straightforward as possible, in order to make it easy for students to grasp and remember.
 A: Do you  mean positive count data? Unbounded?
The negative binomial is popular.
Another good model is the Poisson with inflated 0. That model assumes that either something is happening or it isn't - and if it is, it follows a Poisson. I saw an example recently. Nurses who treated AIDS patients were asked how often they experienced stigmatizing behaviours from others as a result of their involvement with AIDS patients. A large number had never had such experiences, possibly because of where they worked or lived. Of those that did, the number of stigmatizing experiences varied. There were more 0's reported than you would expect from a straight Poisson, basically because a certain proportion of the group under study were simply not in an environment that exposed them to such behaviours.
A mixture of Poisson's would also give you a point process.
A: Number of cigarettes smoked in a period of time: this requires a zero-inflated process (e.g. zero-inflated Poisson or zero-inflated negative binomial) because not everyone smokes cigarettes.
A: Counting processes that aren't Poisson? Well, any finite sample space process like binomial or discrete uniform. You get a Poisson counting process from counting events having independent interarrival times which are exponentially distributed, so a whole host of generalizations fall out of that such as having gamma or lognormal or Weibull distributed interarrival times, or any kind of abstract non-parametric interarrival time distribution.
A: It's unclear if you want counting processes or not. 
If I interpret the 'teaching' tag to mean you are teaching the Poisson process then, for teaching about a process in general, the Bernoulli process is an easy random process to explain and visualize and is related to the Poisson process. The Bernoulli process is the discrete analog so it might be a helpful companion concept. Its just that instead of continuous time we have discrete intervals of time.
An example might be a door to door sales man where we are counting successes by homes that make a purchase.


*

*The number of successes in the first n trials, has a binomial
distribution B(n, p) instead of a Poisson 

*The number of trials needed to get r successes, has a negative binomial distribution NB(r, p)
instead of a gamma distribution

*The number of trials needed to get one success, the waiting time, has a geometric distribution NB(1, p), which is the discrete analog of the exponential.


That's the approach Bertsekas and Tsitsiklis use in Introduction To Probability, 2nd ed., introducing the Bernoulli process before the Poisson process. In their textbook there are more extensions to Bernoulli process that are applicable to the Poisson process such as merging them or partitioning them, as well as problems sets with solutions. 
If you are looking for examples of random processes, and you just want to throw the names out there, there are quite a few. 
The Gaussian process is a significant one in applications. The Weiner process in particular, which is a type of Gaussian process, is also called standard Brownian motion and has applications in finance and physics.
A: As an property/casualty actuary, I deal with real-life examples of discrete processes which are non-Poisson all the time. For high-severity, low-frequency lines of business, the Poisson distribution is ill-suited as it demands a variance-to-mean ratio of 1. The negative binomial distribution, mentioned above, is much more commonly used, and the Delaporte distributions is used in some of the literature, though less often in standard North American actuarial practice.
Why this is so is a deeper question. Is the negative binomial so much better because it represents a Poisson process for which the mean parameter is itself gamma distributed? Or is it because loss occurrences fail independence (as earthquake events do under current theory that the longer one waits for the earth to slip, the more likely it is due to the build-up in pressure), is it non-stationary (the intervals cannot be subdivided into sequences, each of which is stationary, which would allow the use of a non-homogeneous Poisson), and certainly some lines of business allow for simultaneous occurrences (e.g. medical malpractice with multiple doctors covered by the policy).
A: Others have mentioned several examples of point process that are not Poisson.  Because the Poisson corresponds to exponential interarrival times if you pick any interarrival time distribution that is no exponential the resulting point process is not Poisson. AdamO pointed out the Weibull.  You could use gamma, lognormal, or beta as possible choices.  
The Poisson has the property that its mean is equal to its variance.  A point process which has variance greater than the mean is sometimes referred to as overdispersed and if the mean is larger than the variance it is underdispersed.  These terms are used to relate the process to a Poisson.  The negative binomial is often used because it can be overdispersed or underdispersed depending on its parameters.
The Poisson has a variance that is constant.  A point process that fits the Poisson conditions except for not having a constant rate parameter and consequently a time varying mean and variance is called an inhomogeneous Poisson.  
A process with interarrival times exponential but can have multiple events at the arrival time is called a compound Poisson.  Though similar to the Poisson process and having a name with the word Poisson in it, inhomogeneous and compound Poisson processes are different from a Poisson point process.
A: Another interesting example of non-Poisson counting process is represented by the zero-truncated Poisson distribution (ZTPD). ZTPD can fit data concerning the number of languages subjects can speak in physiological conditions. In this instance, Poisson distribution is ill-behaving, because the number of spoken languages is by definition >=1: hence 0 is ruled out a priori. 
A: I believe that you could take your customer-arrival Poisson process and tweak it in two different ways: 1) customer arrivals are measured 24-hours a day, but the store is not actually open all day, and 2) imagine two competing stores with Poisson process customer arrival times and look at the difference between the arrivals at the two stores. (Example #2 is from my understanding of the Springer Handbook of Engineering Statistics, Part A Property 1.4.)
A: You might want to reconsider the soccer example. It seems that the scoring rates for both teams increase as the match goes on, & that they change when teams change their attacking/defending priorities in response to the current score.
Or rather, use it as an example of how simple models can perform surprisingly well, stimulating interest in statistical investigation of some phenomenon, & providing a benchmark for future studies that collect more data to investigate discrepancies & propose elaborations.
Dixon & Robinson (1998), "A Birth Process Model for Association Football Matches", The Statistician, 47, 3.
A: Since the question is related to making the Poisson distribution more understandable, I'll give it a go, since I recently looked into this somewhat for call center incoming call patterns (which follow a memory-less, exponential distribution as time goes on).
I think delving into another tangential model that essentially requires knowledge of Poisson to realize how it isn't one may be somewhat confusing, but that's just me.
I think the trouble with understanding Poisson is the continuous time axis it's on --- as every second goes on, the event is no more likely to occur --- but the further out in the future you go, the more certain it is of happening.
Really, I think it simplifies the understanding if you just trade the 'time' axis for 'trials' or 'events'.
Someone can correct me if this is way off base, as I feel it's an easy explanation, but I think you can replace the flip of a coin, or the toss of a dice, with 'time until a phone call arrives' (what I typically use for Erlang C/ call center staffing).
Instead of 'time until a phone calls arrive' ---- you can replace it with ... 'rolls until a dice hits six'.
That follows the same general logic. The probability (like any gambling) is completely independent every roll (or minute) and is memory-less. However, the likelihood of 'no 6' decreases ever slower but surely towards 0 as you increase number of trials. It's easier if you see both graphs (likelihood of call with time, vs. likelihood of six with rolls).
I don't know if that makes sense --- that's what helped me put it together into concrete terms. Now, the poisson distribution is a count rather than 'time between calls' or 'trials until rolling a six' -- but it relies on this likelihood.
A: Number of visits by an individual customer to the grocery store within a given time interval.
After you have been to the grocery store, you are unlikely to return for a while unless you made a planning mistake.
I think the Negative Binomial distribution could be used here, but it is discrete, whereas the visits are in continuous time.
