How good is the Huffman code when there are no large probability letters? The Huffman code for a probability distribution $p$ is the prefix code with the minimum weighted average codeword length $\sum p_i \ell_i$, where $\ell_i$ is the length of the $i$th codword. It is a well-known theorem that the average length per symbol of the Huffman code is between $H(p)$ and $H(p)+1$, where $H(p) = -\sum_i \, p_i \log_2 p_i$ is the Shannon entropy of the probability distribution. 

The canonical bad example, where the average length exceeds the Shannon entropy by almost 1, is a probability distribution such as $\{.999, .001\}$, where the entropy is nearly 0, and the average codeword length is 1. This gives a gap between the entropy and the codeword length of almost $1$.

But what happens when there is a bound on the largest probability in the probability distribution? Suppose, for example, that all the probabilities are less than $\frac{1}{2}$. The largest gap I could find in this case is for a probability distribution such as $\{.499, .499, .002\}$, where the entropy is slightly more than 1 and the average codeword length is slightly less than 1.5, giving a gap approaching $0.5$. Is this the best you can do? Can you give an upper bound on the gap that is strictly less than 1 for this case?

Now, let's consider the case where all the probabilities are very small. Suppose you choose a probability distribution over $M$ letters, each having probability $1/M$. In this case, the largest gap occurs if you choose $M \approx 2^k \ln 2$. Here, you get a gap of around
$$ \frac{1 + \ln \ln 2 - \ln 2}{\ln 2} \approx 0.08607. $$
Is this the best you can do in the situation where all the probabilities are small?

This question was inspired by this TCS Stackexchange question.
There are lots of papers that study exactly the problem you mention. The first in the series is a paper by Gallager, "Variations on a Theme by Huffman", IEEE-IT, vol. 24, 1978, pp. 668-674. He proves that the difference between the average codeword length of an Huffman code and the entropy  (he calls that quantity "redundancy") is always strictly less than $p$ (= largest probability in the probability distribution), in the case  $p\geq 1/2$, and it is less than $p+0.086$, if $p&lt;1/2$. Better bounds are known, you can find them in the numerous papers that quote Gallager work.