Assuming the query [10^100 planck lengths to lightyears] is a correct interpretation of the question, a tooth on the first gear would need to move ~10^49 light

Assuming the query [10^100 planck lengths to lightyears] is a correct interpretation of the question, a tooth on the first gear would need to move ~10^49 light years (well, in a circle) before a tooth on the last gear moved 1 Planck length. I am less familiar with arc-Plancks, but I assume that is sufficient to say "the last gear doesn't move".