If $d(n)$ denotes the number of divisors of $n \in \mathbb{N}$, we may define the function $$C(n) = \frac{\log(d(n)) \cdot \log(\log n)}{\log 2 \log n}.$$

According to Wikipedia, the Swedish mathematician Carl Severin Wigert proved that $\displaystyle \limsup C(n) = 1.$ $C$ can therefore reasonably be used as a measure of degree of compositeness. However, the natural log base is somewhat arbitrary, and arguably a better choice would be to use logs base 2, $\log_2 n$, and set $$\displaystyle C_b(n) = \frac{\log_2(d(n)) \cdot \log_2(\log_2 n)}{\log_2 n}.$$

This also has $\displaystyle \limsup$ equal to 1, and has the further nice property that $C_b(2) = 0$, so that the minimal degree of compositeness is zero.

Suppose we define an "ultracomposite number" to be an $n \in \mathbb{N}$ such that $C_b(n) > 1$, and for every $m < n$, we have $C_b(m) < C_b(n).$

It follows that the set of ultracomposite numbers is finite. Two questions now arise:

(1) What is the largest ultracomposite number, or failing that, what is a bound or estimate?

(2) How large is the set of ultracomposite numbers, or at least what is a bound or estimate for the size?

(3) More specifically, I conjecture there are seventeen ultracomposite numbers, of which 55440 is the largest.

It should be noted that the interest of this question depends on the acceptance of log base 2 as the privileged choice for this problem--natural logs, for example, lead to far more ultracomposites.