I just watched one of the best episodes of Futurama about nano "gray goo"!
https://futurama.fandom.com/wiki/Benderama
https://en.wikipedia.org/wiki/Benderama
https://en.wikipedia.org/wiki/Gray_goo

BTW the equation is somewhat wrong (thanks gnovice!)
https://scifi.stackexchange.com/questions/4291/what-was-the-equation-shown-in-benderama

but still very interesting, it's an example of
https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)
which is divergent, which means grows to infinity (although very slowly)
see:
https://en.wikipedia.org/wiki/Ant_on_a_rubber_rope
and:
Ant On A Rubber Rope Paradox by Vsauce2
https://www.youtube.com/watch?v=OM9KepKsg6U

IRL (In Real Life) each gray goo machine keeps producing machines, not just multiplying once, plus they are the same 100% size.
i liked very much the twist that Aaron Ehasz the writer of this episode made the nano replicators independent of special raw materials because they make their own from whatever they find (in the episode they make ethanol from water and CO2).

Balls of Steel 2011 - Futurama