[math-fun] Nearest rationals algorithm

I think there was a discussion about 'nearest rationals to pi' ... and I think there's a 'sequence' about that somewhere. I'd be interested in an algorithm to generate such sequences, ... and I'd like to see what results it gives for sqrt(2)^sqrt(2) ... a number which has a specific role in proving that a^b can be rational even if 'a' and 'b' are not [see, e.g., The Princeton Companion] Guy