"Ultracomposite" numbers 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.
 A: What you want to look at are Ramanujan's "superior highly composite" numbers, described, for example, in a 1988 survey in English by Jean-Louis Nicolas. Email we if you wish a copy.The shc numbers can be readily produced, in order, on a computer. I recently improved by software for this type of thing owing to a related question by Igor Rivin. See http://oeis.org/A002201  and http://oeis.org/A002201/b002201.txt Furthermore, since the prime factorizations are figured out before multiplying out the number itself (put in order, the next one is just a small prime times the current one), various ratios of the type you want can be found without needing to print out the number itself. Maybe the best way to convey the advantage of working with these numbers is to point out that no factorization is required; you know the prime factorization before you know the number. 
Anyway, the shc numbers give the biggest ratios of your type. Indeed, they provide bounds for all numbers, such as
$$  d(n) \leq n^{\frac{1.537939860675   \log 2}{\log \log n}}  $$
$$  d(n) \leq n^{ \left( \frac{ \log 2}{\log \log n} \right) \left( 1 + \frac{1.93485096797}{ \log \log n}  \right) }  $$
and
$$  d(n) \leq n^{ \left( \frac{ \log 2}{\log \log n} \right) \left( 1 + \frac{1}{ \log \log n} + \frac{4.762350121}{ (\log \log n)^2}  \right) }  $$
I'm not entirely sure what happens if you drop the 4.76235 term in the third one, I think the conjecture would be that infinitely many shc numbers make the revised inequality false, but the numbers will be really huge. 
Hmmm. Nicolas does not conjecture that, so let's call that a guess on my part. Meanwhile, he says that the Riemann Hypothesis implies that, with
$$  \theta = \frac{\log (3/2)}{\log 2} \approx 0.5849625,  $$ there is a constant $c$ such that (result by Ramanujan) 
$$  \frac{\log d(n)}{\log 2} \leq \operatorname{li} (\log n) + c (\log n)^\theta. $$
He points out that the methods being described would allow one to find the best possible $c$ in this formulation but that is not in this particular survey. If you think about the usual asymptotic expansion for the logarithmic integral function, you see how Ramanujan's RH implication closely resembles the unconditional results above it. 
EEEDDDIIITTT: I made a jpeg of some info I once printed out, the specific s.h.c numbers that give equality in the Nicolas-Robin bounds above. Note that
$$ 1.537939860675 \log 2 \approx 1.066018678297. $$ Given some $\delta > 0,$ the exponent of the prime $p$ in the corresponding s.h.c. number, call it $N_\delta,$ is
$$ \left\lfloor \frac{1}{p^\delta - 1} \right\rfloor. $$ So, in the jpeg, we see the four real coefficients indicated, then the three s.h.c numbers, each with its number of divisors, then a value for $\delta$ that will produce that s.h.c. number written by hand. Note that the three s.h.c. numbers depicted are lines numbered 15, 39, 87 in THIS LIST.

