I think the point here is that there is no "most usefule algorithm ever written" since usefulness is based, by definition, on the situation at hand, so the answer to your question would vary dramatically based on the situation that you need to use the algorithm for.
I did one to do permutations on any number of characters and output it to a text file. I'm talking of hours of data crunching and huge huge text files. It even used about 90% of processor power on my lower end machine. This is the best way I've found to use up space on someone's computer :tongue2:
The implementation of basic operations such as multiplication/division and even addition at the circuit level often use some interesting algorithms for optimization and speed. I'm not sure if you were being serious or not.
This is true of course, since cascading adders are generally not used in today's high speed processors and optimizing synchronous adders can be very tricky, and fine tuned! However, since this is the programming forum, I assumed that the user meant a programming algorithm. In C++, the "algorithm" to add two integers is a=b+c;. And of course, this post wasn't at all serious! ;-)