34 789.
Source(s):
Going through iTunes using the Browse function. Opening all categories picking "All" in Subcategory and then counting the results from all categories.

2 249 + 2 154 + 2 074 + 2 107 + 2 212 + 2 155 + 2 325 + 2 188 + 2 156 + 1 543 + 2 284 + 2 327 + 2 230 + 2 365 + 2 215 + 2 205 = 34 789

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More than 100,000. There is not an exact figure because the number of podcasts is constantly growing.
Source(s):
http://www.apple.com/itunes/whatson/podcasts/

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i came up short after searching for something more recent than the september 2007 figure. found the same article referenced by rocknrolla + an article from sept 2006. the 2006 article counted about 83,000, which is 10x more than the previous year. so, from the beginning in 2005 with 8,000, to 83,000 in 2006, to 125,000 in 2007. my guess is around 225,000. it grew by 10x in the first year, a little less than 2x the following, it probably didn't double in the last year. however, it is really hard to keep track. also, there are a couple of dead podcasts - podcasts that haven't been updated in more than a year.
Source(s):
http://typicalmacuser.com/wordpress/2006/09/24/itunes-podcasts-grow-10-fold...

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