Digital Magnitude Comparator: LSb first or MSb first
There is a sorting algorithm which sorts from the least significant digit, where the size of the digit can be chosen arbitrarily, and is sometimes set to a single bit. This is the only place I can think of off the top of my head where you would wish to compare the least significant bit first, though I'm sure there are other places.

EDIT: Forgot to put the link in there!