I recently noticed some copy stating that something was compatable with "Firefoz 3.6, IE 8 and Google Chrome". I realized I don't know what version my copy of chrome is, yet it's always up to date. Meanwhile I'm painfully aware of which version of Firefox and Internet Explorer I have. Then I realized how much better everything would be if I weren't. I have better things to do.
Google Chrome keeps up to date automatically and, being a web browser, it almost always has an internet connection when in use, so it updates itself all the time. It also does this almost silently; no user interaction is required to install updates, except Chrome must be closed for updates to install. If the user doesn't close Chrome for a while after an update is applied, a little notification tells you that you can update Chrome by restarting the browser. Some consider even that to be too much interaction.
Obviously I can check my exact version number for support/debugging reasons; technically versions are just as real as they ever were. The difference is that the version number only "exists" for technical reasons; versions should not be a barrier to normally interacting with the product.
It's important to note that automatic updating systems that require user input are often extremely annoying and thus often ignored, so I don't consider this the best solution. I'm talking about silent (as feasible), automatic updates.
In the era of Web Apps this seems to be getting more and more comment; after all with a web app the burden of updating is on the website, not you. Mobile operating systems like iOS and their apps seem to be leaning toward automatic updates as well; the Android store now allows you to automatically update applications by default, only asking permission to update an app when the app requires different "permissions" to use things like GPS ect.
Versions are an artifact of development; why should the user be burdened with them? One of the most painful parts about using some software (like a new operating system) is doing updates, depending on the level of user interaction involved.
Should normal users have to care about version numbers in the age of always-on Internet connectivity and Web Apps?