In a previous comment, I sought to further understand how the Lottery possesses the Markov property. Based on your definition above, I can see that it does simply because the distribution X_t of winning numbers has the same dependence on X_{t-1} as it does on X_1, ..., X_{t-1}, that is, zero. Do I have that correct?
For an example without the Markov property, consider the sequence of random variables X_1, X_2, ... with X_1 being either -1 or 1 with equal probability, and X_t being normally distributed with mean X_1 and standard deviation 1.
Knowing the history X_1, X_2, ..., X_{t-1} gives you the exact distribution of X_t (since you know X_1), while only knowing X_{t-1} gives you much less information. This fails to be a Markov chain because the state X_{t-1} doesn't "remember" which of the two possible distributions is being used.
Edit: Wikipedia, of course, has a good overview of applications of Markov chains. http://en.wikipedia.org/wiki/Markov_chain#Applications