Since none of the answers really address your problem (or at least not correctly) I'll give it a go.

The browser caches youtube videos as Flash (.flv) although the extension is not apparent. With Internet Explorer, you can see the cache in Tools -> Internet Options -> Settings -> View Files. With Firefox you can type about:cache into the address bar, and under Disk Cache Device, click List Cache Entries. Viewing the files in Firefox makes it far less cryptic so you can determine the contents easier.

With IE an explorer window will open, you can simply sort the files by size or date modified, and look for files that start like this: videoplayback?ip=0.0.0.0. Use the timestamps to determine if it's the right one. You can simply copy that file to desktop and rename it with the .flv extension.

With Firefox on the disk cache page, press Ctrl + F to bring up the Find dialog and type in videoplayback, you should come to an entry in cache. Check timestamps to determine if the video is the right one, if it is click on it, if not click next in the find dialog until it is. There will be a link on the next page, right click it and select "save target as" and point it to your desktop. It should save instantly since the video is in your cache. Rename the extension to .flv if required. Since this process is a little extensive, here it is step by step:

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As far as programs and add-ons go, I don't think there currently is one for this.