>>484
Of course, you ask to us: "Do you know of any interesting waves to list all possible k-tuples of possible nonnegative integers in order?" In effect it exists, it is sufficient to change the base of natural numbers. e.g. For k-tuple in base two until \begin{equation} f(2^k-1) \end{equation} you have the k-tuple in order  f(0))(0,...,0,0,0), f(1)=(0,...,0,0,1), f(2)=(0,...,0,1,0), f(3)=(0,...,0,1,1), f(4)=(0,...,1,0,0), f(5)=(0,...,1,0,1), etc here you can see that (1,0,0)>(0,1,1) where a>b means a comes after b.
If, otherwise, you fixed k and you need numbers bigger than \begin{math} f(2^k -1) \end{math} you can change base into base n for number until \begin{math} f(n^k -1)\end{math}. e.g. k=2, number until f(125), n≥12 you obtain: (for n=12) f(0)=(0,0), f(1)=(0,1), f(2)=(0,2), f(3)=(0,3),..., f(10)=(0, A), f(11)=(0, B), f(12)=(1,0), f(13)=(1,1), f(14)=(1,2),...,f(125)=(A,5)