it renders them, but most of these "apps" are ActiveX controls that do something in addition to the rendering engine of ie6.
I work in the legal sector in a state of Australia, and our justice system was running on ie6 (last I heard; 24 months ago) and their public website for Barristers (fancy lawyers) to access a system they had invented for easy access to state criminal law, relied exclusively on an Active X control that only ran in ie6.
Thankfully, they have since updated the ActiveX control to work properly in ie7. It wasn't the rendering engine (which I believe that "compatibility button" switches to) that was the issue.
My biggest problem was once a year they would come and make a presentation to our barristers, as I was the IT guy, I'd have to re-install windows XP on a laptop somewhere for use as the presentation machine. I tried a Virtual machine one year, but unfortunately the hardware in the presentation laptop at the time completely choked. In the end, I started saying if they want to present it they need to supply their own laptop.
Sadly, they were presenting it so people being presented to could learn about it for their own criminal trial work. Ironically, one year they basically had to say: "but .. it won't work for you because no computer you have access to will have ie6 on it. And chances are if you own your own computer it won't have ie6 either."
*shrug*.