When is it a good aproximation to consider a star to be an ideal gas? I am currently taking a first course on stellar astrophysics, and I noticed that in some cases we use the ideal gas equation of state for stars, so we also use $\gamma =5/3$. Of course it can only be applied where there is no nuclear reaction, so there is a limit in the temperature.

Also, if the radiation pressure if relevant, you have to consider the parameter $\beta$ to calculate the total pressure and the adiabatic coefficient $\gamma$. It is also incorrect (I think) if there are degeneration or relativistic considerations.

I don't know if I have to consider other factors before I can use this equation of state and value for $\gamma$, maybe it is important if it is a convective or radiative area, or other factors I didn't consider.

My question is: can anybody tell me the limits of the ideal gas approximation? (quantitative better than qualitative but any help will be well received)
There is no hard and fast answer. To be treated as an idea gas (your title question), the particles in your gas should be point-like and they should be non-interacting if you are to use the ideal gas approximation.

This means you can take the mean separation ($\sim n^{-1/3}$) and compare that with the size of the particles - it should be much bigger. You can also compare the particle kinetic energies with the potential energy of their interactions at typical separations. eg The ratio of $kT$ to the Coulomb potential energy should be large.

The rest of your question is confused. You are asking about an adiabatic index, but that is a completely separate question from whether the gas is ideal. It depends on the equation of state, on interactions, on internal degrees of freedom, on the way energy is transported.

Relativistically degenerate gases can be ideal gases and are often assumed to be so. Do not confuse an ideal gas with a perfect gas (even though many do so).