When Huffman coding is inefficient?
If $X$ is any non-constant source, then any codeword in any prefix code for $X$ has length at least $1$, and so $R \geq 1$. Therefore if $X$ is a source with very low entropy then there's a large discrepancy between $H(X) \approx 0$ and $R \approx 1$.

As a concrete example, consider a binary source in which the probability of one of the options is $\epsilon$. Then $H(X) = \epsilon \log (1/\epsilon) + O(\epsilon)$ whereas $R = 1$.