Just a short comment.
If we let f(n,k) be the sum of the divisors of n, each raised to the k power, then the highly composite numbers are those n which are record setters for k=0. For k=1, we get the highly abundant numbers, and for k=-1 we get the superabundant numbers.
A curious question to consider is the set of k values for which a given n is a record setter.
I just calculated that 672 is a record setter for 0 < k < 0.3705405845106956751517917 using Pari/GP. Note that k>0, so 672 is not highly composite, but minimally so (I haven't actually checked that, but it is certainly a record setter down to tiny positive k, yet not for k=0).
Plouffe's Inverter gives me nothing on this 0.3705.. number. It's certainly some solution of a ridiculous exponential sum equation.
Here's more (correct) digits just for fun: 0.3705405845106956751517917005244140444958735486823441966658021494