At what density does helium burning start in a star? I have seen several references that say that helium burning begins in a star once the core temperature reaches $10^8$K (such as here) but nowhere that says what density that corresponds to.  Does anybody know a reference that has this value?  And if it makes a difference, I would like to know the density for quiescent helium burning, not helium flash helium burning.
The short answer: $\sim10^4$ grams per cubic centimeter.

From this webpage, I have a few statistics regarding required mean density and during each phase of fusion (the fusion of lighter elements may happen in the "shells" outside the core).
$$\begin{array}{|c|c|}
\hline \text{Fusion phase}&amp;\text{Mean density (g/cm}^3)\\
\hline \text{Hydrogen} &amp; 5\\
\hline \text{Helium} &amp; 700\\
\hline \text{Carbon} &amp; 200,000\\
\hline \text{Neon} &amp; 4,000,000\\
\hline \text{Oxygen} &amp; 10,000,000\\
\hline \text{Silicon} &amp; 30,000,000\\
\hline
\end{array}$$
This is data for a star of $25M_{\odot}$.

Note, however, that this is mean density in the star, not the density in the core, where fusion is taking place. That is on the order of $\sim10^4$ grams per cubic centimeter (see e.g. here). There may be a sharp density dropoff starting from the center of the star, showing that only the hot, dense inner regions can fuse the main element in each stage.