Perhaps you could add a background image to the print.css file and link this background image to a file on your server that is registering.
For instance:
body {background-image:url(http://www.example.com/printlogger.aspx);}
I'm not sure if this works, just a thought
Update: I just tried this. It increments the counter if you are doing a preview. However, when updating the counter when printing a page (even when disabling background images for printing), this does not happen. Another option would be to use a CSS content property.
UPDATE II . You can use the content property, it works in Firefox and not in IE8. I have not tested other browsers:
body { content:url(http://www.example.com/Count.aspx); }
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