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.