So the question is in the title itself. The minimum number of divisors are $2$ (for primes), but the maximum number of divisors are what we are interested in. It cannot be more than the number itself for sure, so it's bounded above by $n$.
We know that the number of divisors of $\prod_{i=1}^jp_i^{k_i}$, where $p_i$'s are distinct primes is given by $\prod_{i=1}^j(k_i+1)$. We have to maximize this given the constraint that $\prod_{i=1}^jp_i^{k_i}\le n$.