Your question is ill-posed. I have not come across a comparison
dense-denser-densest.
The term "dense" is a topological property of a set:
A set A is dense in a set B, if for all y in B, there is an open
set O of B, such that O and A have nonempty intersection.
The rational numbers are indeed dense in the set of real numbers
with the standard topology. An open set containing a real number
contains always a rational number.
Another way of saying it is that every real number can be
approximated to any precision by rational numbers.
There are denser sets, if you are willing to consider more
elements.
Suppose you construct a set consisting of the rational numbers
plus all algebraic numbers. The set of algebraic numbers is also
countable, but adding them, makes it obviously easier to
approximate real numbers.
Can you perhaps construct a set less dense than the set of
rational numbers?
Suppose we take the set of rational numbers without the element
0. Is this set still dense in the real numbers? Yes, because 0 can
be approximated by 1/n, n>1.
In fact, you can remove finite number of rational numbers from
the set of rational numbers and the resulting set will still be
dense in the set of the real numbers.
