It’s actually kind of tricky to construct a set whose cardinality is a higher order of infinity than a given infinite set. Taking the union of two copies of the set (multiplying by 2) doesn’t work. Taking the union of any finite number of copies (multiplying by any integer) doesn’t work. Taking all ordered pairs, where each element of the pair comes from the set (squaring the number of elements) doesn’t work. Taking ordered triples, or quadruples, etc (raising to any finite power) doesn’t work.
What does work is taking all possible subsets of the given set (raising 2 to the power of the number of elements in the set). That’s why the notation 2^{\aleph_i} has been used upthread. The set of all possible subsets of a set always has a higher cardinality than the original set. If the set is finite, then the cardinality is literally 2^n where n is the cardinality of the set: there are 8 possible subsets (2^3) of a set with 3 elements. It happens to also be true for infinite sets. This is Cantor’s Theorem, and it has a fairly elementary proof.
And although 2^ℵ is most often used, because it’s simplest, if ℵ is infinite, then you can use any finite or infinite cardinal less than or equal to ℵ for the same result.
(In case anyone doesn’t know, ℵ is the Hebrew letter aleph, which is traditionally used for infinite cardinal numbers. There aren’t enough Latin letters for all of the many uses scientists and mathematicians have for them, so other alphabets get drafted for service, too.)
The study of large cardinals, how large can they be, how to get them, is an entire thing is Set Theory.
Here is the thing: the numbers that come out of modelling these physical systems— I will not call them astronomical numbers because those are pathetically too small— like how long it takes for a system to evolve into some weird entropy-defying macroscopic configuration, or for some recurrence to happen, etc., are all finite. Not infinite. So you need some way to construct, estimate, and even describe huge numbers. These recurrence times are, actually, small potatoes in this sense; just a tower of exponentials of a handful of levels will tame all of them.
I saw somewhere a preface to a book on statistical mechanics, that stated that the book will be dealing with three kinds of numbers, ordinary numbers, large numbers, and very large numbers. Large numbers have the property that, when you add an ordinary number to them, they stay the same. And very large numbers have the property that, when you multiply them by an ordinary number, they stay the same.
The number of gas molecules in a tank might reasonably be Avogadro’s number, which is a large number. But the probability that, at any given moment, they will all be on the left side of the tank is 1 over 2 to the power of Avogadro’s number, a very large number.