The reason I'm asking this is that I don't have a circular polarizer yet and I'm planning to buy one.
I know I'll be too lazy to unscrew the UV filter to put on the polarizer. I'm wondering whether I can just screw on the polarizer on top of the UV filter everytime I need it. Has anyone tried to do this? Are there any downsides? Thanks.
You might find that the corners of your image get shaded by stacking two filters, as light from a wide angle that would make it to your sensor normally are blocked by the filter ring of the second filter. This is called mechanical vignetting.
Note that if you are using a lens designed for a full frame camera on a cropped sensor body then it is unlikely to be an issue unless you are using a pretty wide angle lens. Also if you are using a zoom lens and are zoomed in this is very unlikely to affect you.
Other than that you might want to try some test shots with both filters attached and see how the corners of the image cope with the extra filter.
An additional issue is that some light will be lost with every filter. UV filters don't block much light, but they do a little. The polarizing filter will block some more. So in situations where it's borderline whether you can shoot handheld, removing the UV filter might make the difference.
The downsides are:
And, last but not the least, filters get stuck together very easily.
I asked this very same question to one of the managers at a local Henry's store.
He explained that by stacking multiple filters, there is a greater possibility for light to bounce around between the glass and thus cause a ghosting effect (ultimately reducing the contrast of the image). Although most filters are coated to prevent this from happening, they're not perfect. Generally though, the only time you really need to worry about this is if your lens is pointing directly at the sun or a bright light.
One other downside he mentioned is that if you tighten the filters too tight, you might not be able to take them apart afterwards. So be sure to tighten the polarizing filter just enough so that you can turn the dial without causing the filter to come loose.