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So the short answer: there is no easy way that I know of.
Alternatively, if you develop the feed in a scripting language like ASP or PHP, just embed a counter script within the ASP/PHP that outputs the RSS.
i don't think this is the solution that will work for me. I am doing news syndication and the pages are available to public view even without access through RSS.
I have noticed that newsgotor leaves a browser id name the log file including the number of subscribers.
NewsGatorOnline/2.0 (http://www.newsgator.com; 19 subscribers)"
this would be nice if other services would practice the same...
At the moment, unfortunatly, the most part of newsreaders use a simple User Agent IE or Mozilla like.
This make log reading really hard.
I usually provide RSS content with php or asp script and, before sending the content to the client, I log all information I need, as asp4bunnies suggested before.
Feedburner, though, does a really good job of counting -- it checks user-agents and IP-addresses and tries to separate it all out.
On the other hand, sometimes you just have to let go and say: I put this out there on the web, and some people are reading it. Counting the exact number doesn't really make a difference.
Unless you're CNN, though, it's not as vital if your latest item on the RSS feed gets delayed by a few hours. It will just show up in the user's aggregator a few hours later. No lost sales.