James M. answered • 03/24/19

Math nerd interested in furthering math education

So, because there are an infinite number of primes, there are an infinite number of highly composite numbers. A highly composite number's prime factors are made up of the smallest consecutive primes with decreasing powers. For example, 5040 is a highly composite number because it has the factors 2^{4}*3^{2}*5*7. These primes are the smallest consecutive primes, and the powers they are raised to decrease (you can have 1 as a power repeating and it will still work as long as it never increases after you get to 1). Using these rules, we can take any number of consecutive primes and so long as the powers never increase we will have a highly composite number.

You are correct that highly composite numbers will be very large eventually, but there are an infinite number of them.