This is not possible at all. Browsers are separate from the Desktop and code run inside of one shouldn't be able to interact with anything outside of the browser.
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106answers:
4This is not possible "currently", but there's no technical reason why a browser couldn't provide a proprietary API for this, using non standard html/css/js.
However, that's what it would take, a browser to actually implement this functionality and then expose it as an API, and even then it would be browser specific.
UPDATE (as some people have perhaps misunderstood my answer???):
I'm giving technical context to the question. Of course noone's ever going to implement this, but I'm saying it's technically possible.
Also, doing this would not break the sandbox model. The browser itself (forget an API for a second) could implement transparency any way it wanted. Once it that it could hook it up to it's Javascript engine, and create a stupid call: Chrome.Element("").WeirdTransparency()
UPDATE to Questioner's Update:
to your point:
The reason why would I like it is that I want to build a cross-platform (Linux, Windows, Mac), zero-install, fancy-looking rich client application (not meant to be served as an Internet website)
AIR kinda covers 90% or your requirements. It still needs a small install, but apart from that, you're running...
I'm sure browser developers would need a lot of "inspiration" - aka $$$ to do this. It's currently not a feature that a whole lot of people are looking for.
Since standard compliance is very high on the priority list for all browser developers, making this out of the box would be a problem. Namely because there is no CSS/HTML support for it, and the standard is to have a white background. This means that they would need a custom "flag" somewhere in the markup to tell it to switch off the white background.
This would be exclusive to the browser that implements the "feature" and anyone else using any other browser would not be privy to the it.
You mean like this?
Check out screenlets - http://www.screenlets.org/
also check out dashboard widgets (mac), google desktop gadgets (mac/win/linux), and yahoo widgets (mac/win).