In probability, a martingale is given by a sequence of integrable 
  random variables (S_n) and an increasing sequence of 
  $\sigma$-algebras ${\cal F}_n$ such that
  $S_n$ is ${\cal F}_n$-measurable and 
  $E(S_{n+1} \mid {\cal F}_n) = S_{n}$.

  This is an important notion because there are many results concerning
  convergence of martingales sequences, e.g. if it is bounded in $L^2$
  then it converges in $L^2$ norm and $a.e.$

  If $X_i$ is a sequence of i.i.d. random variables and 
  ${\cal F}_n = \sigma(X_i, i\leq n)$, then the following sequences 
  are martingales:

  -   $S_n - E(S_n)$, 

  -   $exp(S_n)/E(exp(S_n))$, 

  -   $(S_n)^2-E(S_n^2)$, 

These are used in the theory of random walks to compute e.g. the mean time before reaching a given state.
  
Are there any other interesting examples of martingales?