What is the maximum possible length of an integer less than 600?

That is not how to correctly define the "length" of an integer. The length of an integer is the number of primes, not necessarily distinct, that you'd need to multiply together to produce that integer. As they've worded the question, the answer would be four, because no positive integer less than 600 has more than four prime factors in total (2*3*5*7 = 210, for example, has four, but the smallest number with five prime factors is 2*3*5*7*11, which is too big).

But using the correct definition of "length", we'll get integers that are both "long" and small by multiplying many of the smallest primes. The number 2^9 = 512, so there is one number less than 600 of length nine. If we multiply ten primes together, the smallest number we could possibly make is 2^10, which is too big, so nine is the answer.

I'm assuming above that we're discussing only positive integers. That's another issue with the question - it needs to say "positive integer". If negative numbers were allowed, we might be talking about a number like -3^1000, and some number theorists would say that has a length of 1000 (because if you're extending prime factorization concepts to negatives, you ignore units like -1).