This Numberphile video, featuring Dr James Grime, is about so-called Highly Composite Numbers.
Or as I prefer to call them, Anti-Primes, because to me they seem as far removed from being prime as possible.
Here - posted on Numberphile's extras channel - is more footage about there being Infinite Anti-Primes and a nice blunder by Ramanujan.
And finally I thought this was worth sharing because:
a) I have seen other people make the incorrect assumption in YouTube comments.
b) It is fun to see how Numberphile videos are made and how we sometimes think on our feet, with mixed results!
If you like seeing the extra footage from Numberphile videos, can I recommend you subscribe to Numberphile2 on YouTube.
Highly Composite Numbers on Wikipedia.
And here is a good resource about very large HCNs.
After filming, I learned I'm not the first person to suggest calling these numbers anti-prime - see this 2009 blog post.