Archimedes's work The Sand Reckoner (also known by its original Greek title Ψαμμίτης or by the Latin translation thereof, Arenarius) sets out to prove that the number of grains of sand in the Universe is finite by providing an upper bound. From my point of view, this text is remarkable because not only it introduces a kind of scientific notation for numbers but also for the idea of putting an upper bound on the order of magnitude of something we cannot directly count (Archimedes arrives at an upper bound of $10^{63}$), and for the introduction of truly large numbers: Archimedes considers numbers like $10^{8\cdot 10^8}$ and $10^{8\cdot 10^{16}}$ (according to Wikipedia — I didn't check) which I presume were far beyond any finite quantity explicitly described by anyone thitherto.
Of course, now, truly huge numbers have been constructed and used for various reasons in mathematics: some have even proposed the word "Googology" for this art(?).
So my question would be about the history of "Googology" and what the main steps after Archimedes were in the description of large numbers. Specifically: when was the first time in history that a number greater than $10^{8\cdot 10^{16}}$ was explicitly mentioned (that is sufficiently precisely described so that it can be compared)? But generally speaking, any comments on the history of large numbers interest me.