"Dense" regular expressions generate $\Sigma^*$?
Your conjecture is disproved by Keith Ellul, Bryan Krawetz, Jeffrey Shallit and Ming-wei Wang in their paper "Regular Expressions: New Results and Open Problems". While the paper is not available on-line, a talk is.

In the paper, they define the measure $|\mathrm{alph}(R)|$, which is the number of symbols in $R$, not counting $\epsilon$ or $\emptyset$. However, $\emptyset$ can be eliminated from every expression not generating the empty language, and the expression can be "cleaned up" so that the number of $\epsilon$ it contains is at most $|\mathrm{alph}(R)|$ (Lemma on page 10 of the talk).

In page 51, for every $n \geq 3$ they construct a regular expression of size $O(n)$ over $\{0,1\}$ which generates all strings of size at most $\Omega(2^n n)$, but doesn't generate all strings. Note that "size" here is both in your sense and theirs, since we're using big-O notation. They also pose an open question to find the best dependence between the two parameters.