What is the X in X-mas?
An algorithm that returns a sequence of approximations to the correct answer such that each approximation is no worse than the previous one, i.e. the algorithm can be stopped at _any time_.
Newton-Raphson iteration applied to finding the square root of a number b is another example:
x = (x + b / x) / 2
Each new x is closer to the square root than the previous one.
Applications might include a real-time control system or a chess program that is allowed a fixed thinking time.