How long can this pattern continue? My guess is forever, since you can just multiply a prime by the next prime (2x3=6), then take that prime, and multiply it by the next (3x5=15) and so on. So it seems to me like there's an infinite number of these kinds of primes, unless I'm not understanding you correctly.
I guess they mean a naming pattern: like hemi-demi-semi-primes
(That's how we do it in music, over on my side of the 'pond': but we get to hemi-demi-semi-quavers then (afaik) give up because no one cares for more!)
I think numbers like these would be very useful in cryptography, where the higher the power of a prime factor sometimes, the easier it is to factor it. Some probably obscure probability calculations and so on