I'd like to respectfully disagree with Jeff on this one. There are plenty of valid reasons to change the file extension. There are no technically valid reasons to do it : it all works in the same way and using file extensions allows the operating system to serve up content more efficiently by determining the type of content from the file extension.
However, there are plenty of other reasons to remove the file extensions, including aesthetic reasons (it just looks better, especially in other media such as printed and visual media). It also is more understandable for normal people to understand a url that hasn't got a file extension on it : the main reason a standard Windows install actually hides the file extensions - they just confuse normal, non technical people. There's a reason why Twitter, StackOverflow and plenty of other sites don't use extensions, because people like extension less urls.
The performance impact of wildcard extensions in IIS6 is vastly overstated for the majority of websites, who don't have enough traffic to really worry the server anyway. If your server is getting enough traffic to start worrying about performance, then you should start investigating this degree of optimisation. For the rest of the sites on the internet, it's more of a challenge to find the traffic rather than worry about CPU cycles serving it up. My personal view is that hardware is cheap and visitors are not. You've paid for that CPU and memory : put it to work. I've assisted a lot of people remove the extensions from their DNN sites and I'm yet to have one person go back to extensions citing poor performance.
With IIS7, the use of extensionless urls becomes a lot easier and IIS will actually optimised for static files and reduce the amount of asp.net processing that occurs. Best of all, you can change the handling of extensions within the web.config file, so you can change it without even telling your hosting company that you have done so, as many of them have an irrational fear of removing extensions.
To answer the original question, no, you can't remove the extensions from a DNN installation without using a third party Friendly Url Provider. The use of '.aspx' is hardcoded within the standard DNN Friendly Url Provider, so you won't be able to eliminate it unless you swap out the standard provider for something else. Of course, I would recommend using the Url Master module, but that's out of self interest! After all, I started out with a question the same as yours: 'can't I get rid of these .aspx extensions?' and went from there.