The number with the most factors is infinite. The number with the most known factors contains all of the prime numbers, of which Euclid's theorem also suggests there is an infinite number.
For the number that has the most factors within a certain number set, keep multiplying increasing prime numbers until the product exceeds the upper limit of the set. Get the product of the prime numbers just prior to the point that the product exceeded the upper limit. This number has the most prime factors in a set. Then keep multiplying this number by 2, the lowest prime number, to increase the number of factors until it reaches as close to the upper limit without going over.