At what point do car speakers need an amplifier to drive them?
When the volume on your head unit won't make them loud enough for your taste without creating audible distortion.
When have you ever seen a speaker that didn't need an amplifier, there is an amplifier in the deck.
All speakers need an amplifier. The amplifier and speakers need to match Ohm wise and the wattage needs to be close (as long as the max wattage of the amp does not exceed the wattage of the speakers).
30 watts +
Average cars today power about 60-80 watts in the listening area without an extra amplifier ( 15-20 rms watts per channel) from the head unit. It takes much more power to have bass sound as loud as treble. Some premium systems provide much more than that and several speakers that can support it. You should not overpower any speakers you have. The point where your speakers need an amplifier is when that much more power is required to be able to hear the full range of sound in the program and not have any of that overcome by noise the car makes, without producing distortion. We can give you formulas all day, but the one thing that will actually give you the idea is if you go to a stereo shop and listen to displayed sound systems. That's the best way for you. Always remember, somebody will always call out a bigger number than the next guy without respect to frequency response.
When you can't hear them.