Based on the source, it's kind of clever. Each "cow" and "haystack" are a pair of radio buttons, so clicking on the cow toggles the select state for both simultaneously. This combines with some fancy CSS animation to make the cows fly around so they're harder to click.
The counter is a large image with every possible score (http://ndruger.lolipop.jp//hatena/20110429/css_game/images/c...). Initially only the top of the image is shown, but as the radio buttons are clicked it makes visible some spacing elements which move the image so the next score is shown.
As the discussion from that thread concluded, it all comes down to how you define "turing completeness" with respect to the status of turing machines as necessarily abstract constructs, what with their requirement of infinite time/space. It also depends how you view the need for human interaction to drive computation as being relevant to the definition.
Neither of these things were really central ideas in turing's original work, so its kinda open to interpretation.
I've seen this, in situations where a link is set to become bold when the mouse pointer is hovering over it. The result is that the text becomes bold, and redistributes itself. Having redistributed itself, the bold link is no longer under your mouse pointer, so the link shrinks back to normal font weight. Now the link is back under your mouse pointer so it becomes bold again. Repeat.
