My favourite "big but comprehensible generation number" is the count of unique shuffles of a standard deck of 52 cards.
(this is my best recollection of the "this is how insane it is" story, numbers may be slightly off)
Iterate through a billion possible shuffles a second and every yearly anniversary, take a 1M pace around the equator. When you get back to your starting point (after 40,075 years), take 1cc out of the Pacific Ocean.
When the ocean is empty, put one A4 piece of 0.05mm paper on a pile. Now you start filling the ocean 1cc every circumnavigation. When it's full, A4 piece of paper on the pile. etc.
By the time you exhaust the list of shuffles, the pile will be 188 light years tall.
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