I’ve seen this video where GOOGOL gear is depicted (it takes 10100 years on reaching the final gear) but how many generations will it take? The 5th gear takes about 10 hours in real time, by the time the 10th one starts spinning (you’re probably already dead).
The entire thing consists of 100 gears. However, even if you record or livestream a video of the entire thing: would you still be alive by the time the final gear starts spinning? The common life span of a human is around 72-73 median but some can reach 100+ up to 125.
Starts spinning? If they’re all connected, technically the moment the quickest gear turns, they all turn at the same time, just each more slowly than the next.
That said, the outer edge of the last gear is being moved by the mechanism at less than a Planck length per second, so it’s actually being moved more by the environment than it is by anything else, so in a way you could say it isn’t moving at all. But it still differs from a gear that is not in the mechanism at simply because it is intrinsically linked to everything else in a very specific way.
Now, if by “starts spinning” you mean “appears to have moved”, that’s a subjective assessment. Eyesight will play an important role. Some people might be able to discern a thousandth of a degree of difference between a fixed mark and a mark on the outer edge of a gear. Are we allowed to use external magnification? What about a scanning electron microscope?
There’s a version of this where the last gear is embedded into a cement block that is bolted to the base of the machine.