I suggest ℵ1.
It’s at least 2 to the power of ℵ0 – the number of factors in the title. And by definition, 2 to the power of some aleph is the next aleph. As every finite set with N elements also has its power set of 2^N elements.
This is the lower bound.
The next reason is, that countably many sided cuboid 1 by 2 by 3 by 4 … has a bijective mapping between the real numbers on the interval [0,1] and its consisting hypercubes 1 by 1 by 1 …
And this is also the upper bound.
ℵ1 is therefore the most natural definition for such a product. That the product of all naturals, is equal to the number of all reals.