According to Wikipedia, in April 2008, the number of videos on Youtube was 83.4 million (ref: http://en.wikipedia.org/wiki/YouTube#cite_note-5). However, the link in the cite note now displays â€œ*â€ video results 1 – 20 of millions, without showing the real count.
Here's one way I found to get an estimated, but relatively accurate, number of videos on the popular video sharing site Youtube. The idea is simple. Get this feed: http://gdata.youtube.com/feeds/api/videos/-/* and parse out the number inside the <opensearch:totalresults> tag.