Here is a prototype for a possible new game mode for Button Masher: Oscillation.
The definition of oscillation according to Google is "movement back and forth at a regular speed." This is exactly what this game mode does. The button moves around back and forth, and you have to move your cursor around to keep up with the button while clicking at the same time.
Oscillation mode works on IE10+, Firefox 16+, Chrome, Safari 4+, and Opera 12.10+.
Update (2020): I wasn't able to port the oscillation prototype for the 2020 site overhaul. You can, however, find it archived on the old site.