Which is greater?

We note that:

$\begin{aligned} \log_{10} 300! & = \sum_{n=1}^{300} \log_{10} n \\ & = \sum_{n=1}^{9} \log_{10} n + \sum_{n=10}^{99} \log_{10} n + \sum_{n=100}^{300} \log_{10} n \\ & > 9 \times \log_{10}1 + 100 \times \log_{10} 10 + 201 \times \log_{10} 100 = 0 + 100 + 402 = 502 \end{aligned}$

$\log_{10} 100^{100} = 100 \log_{10} 100 = 200$

$\implies \boxed{300! > 100^{100}}$

It is easy to see that:

300 > 100, 299 > 100, 298 > 100, ..., 201 > 100 such that 300 299 298 ... 201 > 100^100 from which the inequality 300! > 100^100 can easily be deduced.