All I think can of is the following:
Since the reaminder become strictly smaller each step, the worst case is when the remainders go down by 1 each step, in which case the process will take "b" steps in total to complete. Therefore, the process will terminate in at most "b" steps.
But how can we prove that all reaminders satisfy the inequality r(i+2) < r(i)/2? By induction? or...? This seems to provide a much better upper bound on the number of steps to complete the process.
Any help is much appreciated!
[also under discussion in s.o.s. math board]