Fat Tail / Large Kurtosis Discrete Distributions? All,
I'm wondering if there are any notable, basic discrete probability distributions with "fat/heavy tails" or a large kurtosis?  I know the Geometric Distribution's excess kurtosis approaches 6, but I can't find any that are larger.
Are there any discrete distributions where the kurtosis ( or other higher normalized moment ) is infinite / undefined?
( By basic, I guess I mean something one could easily implement with game tokens like dice or cards. )
 A: Here is an interesting family of discrete distributions that answers Charles' question, and additionally sheds light on what kurtosis is and is not.
Let $X = \mu + \sigma Z$, where $Z$ has the discrete distribution
$Z = -0.5$,   with probability (wp) $.25$
$= +0.5$,   wp $.25$
$= -1.2$,   wp $.25 - \theta/2$
$= +1.2$,   wp $.25 - \theta/2$
$= -\sqrt{0.155/\theta + 1.44}$,  wp $\theta/2$
$= +\sqrt{0.155/\theta + 1.44}$,  wp $\theta/2$.
The family of distributions of $X$ is indexed by three parameters: $\mu$, $\sigma$, and $\theta$, with ranges $(-\infty, +\infty)$, $(0, +\infty)$ and $(0,.5)$.
In this family, $E(X) = \mu$, $Var(X) = \sigma^2$, and the kurtosis of $X$ is as follows:
kurtosis $= E(Z^4) = .5^4 * .5 + 1.2^4 * (.5 - \theta) + (0.155/\theta + 1.44)^2 * \theta$. 
Within this family,
(i) kurtosis tends to $\infty$ as $\theta \rightarrow 0$, providing an example of what Charles wanted. 
(ii) the distribution within the "shoulders" (i.e., within the $\mu \pm \sigma$ range) is constant for all values of kurtosis; it is simply the two points $\mu \pm \sigma/2$, wp $0.25$ each. This provides a counterexample to one interpretation of kurtosis, which states that larger kurtosis implies movement of mass away from the shoulders, simultaneously into the range between the shoulders and into the tails.
(iii) the "peak" of the distribution is also constant for all value of kurtosis; again, it is simply the two points $\mu \pm \sigma/2$, wp $0.25$ each. This provides a counterexample to the often given but obviously incorrect interpretation that larger kurtosis implies a more "peaked" distribution. 
(iv) In this family, the central portion of the distribution actually becomes flatter as kurtosis increases, since the probabilities on  $\mu \pm 1.2\sigma$ and $\mu \pm 0.5\sigma$ converge to the same value, $0.25$, as the kurtosis increases. This provides a counterexample to the often-stated interpretation that higher kurtosis corresponds to "peakedness" and lower kurtosis corresponds to "flatness." Within this given family of distributions, higher kurtosis actually corresponds to a flatter peak.
A: For a "fat tail" you might take 
probability mass function $p(x) = \zeta(s)^{-1}/x^s$ for positive integers $x$, where $s > 1$.  This has finite variance for $s > 3$, and finite kurtosis for $s > 5$.  As $s \to 5+$ the excess kurtosis is, according to Maple, 
$ \dfrac{8100 \zeta(5)}{8100 \zeta(3) \zeta(5) - \pi^8} (s - 5)^{-1} + O(1)$, where 
$8100 \zeta(5)/(8100 \zeta(3) \zeta(5) - \pi^8) \approx 13.82154183$.
For something you can implement easily with a fair coin, you might try this.  Let $N$ be the number of flips of the coin until the first heads, and $X = r^N$.
For the kurtosis to be finite, you need $r < 2^{1/4}$.  According to Maple, the excess
kurtosis is $${\frac {-13\,{r}^{7}+16\,{r}^{5}+2\,{r}^{4}+8\,{r}^{3}+16\,{r}^{2
}-16}{2 \left(2- {r}^{4}\right)  \left(2- {r}^{3} \right) }}$$
For example, if $r=1.1$ the excess kurtosis is  approximately 24.22623314.
