At what point does Avogadro's law not apply?
My feeling is that the size of the molecule matters much less than you would think. For instance, from a purely ideal gas perspective (as Avogadro's law assumes), the expected value for the speed of a particle is $$v=\sqrt{\frac{8kT}{\pi m}}$$
where $k$ is Boltzmann's constant, $T$ is the temperature, and $m$ is the mass of the particle.

So, that means that as the mass of your particle doubles, the average speed of the particle is decreased by a factor of $1/\sqrt{2}$. But, since pressure, from a kinetic molecular theory perspective treats pressure as being the average force which the particles apply to a wall (over a certain area), and force is the time derivative of momentum $p=mv$, we see that $p$ increases by a factor of $\sqrt{2}$ when the mass is doubled (because $v$ decreased by $1/\sqrt{2}$.) That means, the average force on the wall will be greater per collision, but not very much greater.

For instance, I went ahead and calculated the pressure which a mole of hexane exerts on $22.42\ \mathrm{L}$ according to the van der Waals (VDW) equation and found it was $1.05\ \mathrm{atm}$. So, as expected, the value is larger, but not much larger. That's because VDW actually accounts for a parameter (b) which is the volume excluded per mole (meaning the volume that particle actually takes up, sort of).

So, hexane, which is significantly larger than $\ce{CO2}$ or anything else mentioned, only increases pressure by 5% when confined to the same volume. That also means the volume should be 5% larger.

Be wary of scaling that to larger molecules though! As hydrocarbons get larger (and being linear helps) they have a much greater polarizable volume which results in (how about this for a name) greater van der Waals attractions. This extra attraction counteracts the increase in size of the molecule a pretty significant amount.

All that to say, the deviation from Avogadro's Law at SATP is actually quite small.

You're only going to get noticeable deviations at SATP when you have very polar molecules, even then, however, it's not too bad. Water for instance takes up $22.45\ \mathrm{L}$ at SATP which isn't even deviating in the direction we would expect, but if you increase the pressure to $1.1\ \mathrm{bar}$, then the volume occupied for water is $22.33\ \mathrm{L}$.

Intermolecular forces are more significant than the particle's size, particularly at lower temperatures and higher pressures.