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?

You would need to spin the first gear 1 trillion times per second from now until the last proton in the universe decays, then do that about 10^68^ more times. 10^68^ is roughly equal to 52!, the number of unique ways a deck of cards can be shuffled. There's a somewhat famous description of how large this number is (paraphrased for brevity):

Start a timer that will count down the number of seconds from 52! to 0. Every billion years, take a step along the equator. Every time you go around the world, remove one drop of water from the Pacific Ocean. Every time you empty the ocean, take one sheet of paper and place it flat on the ground. Every time the stack of paper reaches from the Earth to the Sun, take a step up Mount Everest. Once you've reached the summit 80 times, the timer will have reached 0.

10^100 is an absurdly long time. We don't know how long future generations will be. Humans may become effectively immortal due to advances in healthcare and life extension technology.

Or we might become extinct. Or we might become the former first, then become the latter. Or we might become ephemeral magical energy-beings with control over space and time and dimensions never before dreamed of. The entire universe may be extinct, nevermind us. It may have collapsed into a state devoid of entropy, or it may have exploded into an arrangement we can't even comprehend.

And that's not even talking about whether the gears will still exist. Even in the foreseeable near future, we almost immediately will have a "Ship of Theseus" problem. Assuming the gears are made of any currently known or hypothetical material, it doesn't matter what it is or how you lubricate them, all the teeth will have worn completely flat on the first gears long before the backlash is even accounted for on the final gears so they can even start to move.

So who's maintaining and replacing these gears for all these generations? And how can you trust that they will? How do you know somebody's not going to decide to reduce the number of gears (because, you know, budget cuts), and what if it keeps happening over the generations and millennia? What if somebody slips while repairing it and the teeth of the final gear come loose and start spinning. Or what if gasp somebody decides to intentionally spin the final gear by hand? Or what if somebody decides to throw the whole set of gears into the trash can? What if the building they're in burns down? People are untrustworthy. Generations of future people cannot be trusted to continue and maintain your experiment indefinitely, and it's far more likely that the gears will be discarded, repurposed or thrown away than it is that they will be maintained long enough for the final gear to start turning.

The fact that you are even asking this question suggests you don't really understand how many years 10^100 years really is, it's obscenely large in both human and physics timescales. It is so far outside our realm of possible comprehension and understanding that us even trying to reason about it is at least quadrillions of times more ridiculous than asking a virus to to debate morality and ethics. It is at least quadrillions of times more than our entire civilization has existed. It is quadrillions of times more than our planet has existed. It is a timeframe impossible to coherently reason about. That's the whole point.

A "generation" isn't really a well-defined unit of measure. Generally, it's thought to be about 30 years for a given group, but it can vary; I've seen some define it as 40 years. If you mean lifetimes, that's going to vary by a lot of factors.

But to your second question, someone in the comments did the math, and I think it also answers your core question:

• The first gear takes about 3.5 seconds to turn.
• The second gear takes about 35 seconds, or 3.5x10^1.
• The fifth wheel will take around ten hours to turn once.
• In a month, the seventh wheel will have almost one rotation.
• The eighth will take a little over a year.
• If you watch this machine from the time you are born until the time you die, you will probably live to see the tenth gear make most of one rotation.
• The eleventh will take over a millennium to turn.
• The twelfth considerably longer than all of recorded history.
• The fourteenth wheel would take about as long as humans have existed.
• In the time since the dinosaurs went extinct, the sixteenth wheel would turn a little more than half way.
• Earth's existence has been long enough to get the eighteenth wheel half way around.
• In the entire history of the known universe, the twenty-first gear would move by just over one tooth.

ETA: all of these calculations are based on that initial 3.5sec rotation. If you increase RPM, obviously the machine will turn faster accordingly. There's some motors that can spin at 10k, 250k, and even 1M RPM.