An upper bound for $\log \operatorname{rad}(n!)$

$\text{rad}(n!)$ is just the product of the primes less than or equal to $n$ (so a primorial), so

$$\log \text{ rad}(n!) = \mathop{\sum_{p \le n\;|\;p \text{ prime}}} \log p$$

also known as the first Chebyshev function $\vartheta(n)$. The statement that $\vartheta(n) \sim n$ is equivalent to the prime number theorem, and the Riemann hypothesis gives bounds on the error of this approximation. More information is available at the Wikipedia article and in most books on analytic number theory.

Erdős' proof of Bertrand's postulate includes as a lemma the elementary upper bound

$$\vartheta(n) \le n \log 4.$$

A short proof: Notice that all the primes $p$ with $n<p\leq 2n$ must divide $\binom{2n}{n}$ so that $\frac{\text{rad}((2n)!)}{\text{rad}(n!)}\leq \binom{2n}{n}$. Also, $\binom{2n}{n}\leq 4^n$. Consequently for $n=2^k$

$$\text{rad}(n!)\leq \binom{n}{n/2}\cdot \binom{n/2}{n/4}\cdots \binom{2}{1}\leq 4^\frac{n}{2} \cdot 4^\frac{n}{4}\cdots 2\leq 4^n$$

and we see $\log \text{rad}(n!)\leq n\log 4$ for $n=2^k$. You can get other $n$ by noticing $\log \text{rad}(n!)$ is monotonic. (Without cleverness you could lose a factor of $2$ at worst)