There's natural density, which is a notion of size for subsets of the naturals that can differentiate between subsets of equal cardinality.
There are also ordinals, which are much finer than cardinals but have the disadvantage that they only apply to well-ordered sets, and ordinal arithmetic works very different to the naturals.
And then a final one - exotic models of the real numbers like the surreals contain infinite quantities of different sizes that can be compared, divided, added just as you like and the order relation works intuitively. The disadvantage here is that they are generally harder to define and reason about than the ordinary reals. The reason people like them is that you they contain infinitesmals too, which you can use to formalise an alternative foundation for calculus.
(although strictly speaking this last one isn't about sets and sizes, it's in a similar cluster of ideas about infinite arithmetic)
There are also ordinals, which are much finer than cardinals but have the disadvantage that they only apply to well-ordered sets, and ordinal arithmetic works very different to the naturals.
And then a final one - exotic models of the real numbers like the surreals contain infinite quantities of different sizes that can be compared, divided, added just as you like and the order relation works intuitively. The disadvantage here is that they are generally harder to define and reason about than the ordinary reals. The reason people like them is that you they contain infinitesmals too, which you can use to formalise an alternative foundation for calculus.
(although strictly speaking this last one isn't about sets and sizes, it's in a similar cluster of ideas about infinite arithmetic)