The problem, as it has been for many people, is a Microsoft "improvement" to Windows 7's graphics to make it more Windows-8-like and support IE10 - but it also breaks ATI's display drivers. This is the culprit: update KB2670838. I have an ATI Radeon HD2600XT and this has been vexing me for months. It has also been making Windows Calculator unusable - which is where I found the solution (http://answers.microsoft.com/en-us/windows/forum/window...).
Shame on Microsoft for releasing a 'patch' which breaks so many people's display drivers - and not fixing it!
As soon as I uninstalled this patch (via Control Panel / Programs and Features / Show Updates) and rebooted, CCC began working just fine - and Calculator was back to normal too.