Level 1: Truly, Undeniably Impossible
Even with all of the amazing things web developers can do these days we still have one solid restriction: the browser. A browser instance cannot affect other browser instance (unless they are linked). One domain cannot directly read another domain's cookies. Web sites cannot affect a user's computer unless the user allows it. This level of impossibility is usually not reached, since most clients want to duplicate web functionality that they've seen on the web instead of invent brand new ones.
Level 2: Sensibly Impossible
Level 3: Morally Impossible
There's a reason why some generic practices are labeled 'best'. Moving styles to an external sheet, limiting the number of links in a navigation bar, or optimizing content pages for SEO rather than layout styles. Some of these are mostly internal items that most clients do not concern with, but I've had requests that violate basic SEO and security guidelines (cross-domain scripts and iframes, for example). Client requests like this can turn into great discussion starters for updating other portions of a website to a more compliant and 'best practices' state.
Level 4: Internally Impossible
Diving into different frameworks (and building my own) has given me some great experience with impossible updates. Frameworks have a horrific way of limiting developers. By narrowing down the scope of code for manageability, possibilities can easily become limited and unreasonable. Depending on how much time it will take to retool a framework and how much it will break internal practices (which may or may not be related to how important the extra functionality is), this level of impossiblity is solely of the web developer/department's creation.
Level 5: Personally Impossible
Similar to the last level, this is solely up to a web developer. One thing I do with my sites is try to limit the number and size of http requests for a website. Using sprite images, combining stylesheets and scripts, and keeping the dom minimal are all very important to me. It's not impossible to include separate styles and scripts to implement Facebook API logic into a website but it does raise overhead for loading times. I try to take these problems individually, weighing the pros and cons of each one, before adding something to a site that violates my best practices. If the cons are too much than it is impossible for the scope of the project.
As I stated before, only a few things are truly impossible. The majority of client requests that web developers trip over are based on a lack of understanding of web technologies and best practices. By explaining these ideas to the client instead of either just doing it (and feeling guilty later on about breaking best practices or internal guidelines) or simply saying 'No', a web developer should take the opporuntity to educate the client and find an acceptable solution that meets the demands and keeps the project sensible.