How can I extend the ten minute video limit on a Canon PowerShot ELPH 300 HS?
I believe the limit is due to the file size, not the actual recording length.  This forum thread seems to confirm this for the T2i/500D, and Wikipedia agrees for the 5D mkII.  Also according to these sources, the limit for both cameras is 4gb.

For the PowerShot cameras, the file size limit is likely much lower. For the S2, it appears to be 1gb.

What all this means is that the time limit isn't a time limit, per se... but a file size limit.  So if you are able/willing to reduce the file quality, you may be able to extend the time limit.

If you can shoot in a lower quality video mode (say 480i instead of 720p), that will likely improve your recording time.  If you can reduce the frame rate, that may improve your time limit, as well.  Of course all of these things will make the video noticibly lower quality, as well.

If you can't get different equipment, you may be best off strategically stopping and starting the video during pauses in the action.