Can a Kage Bunshin make another Kage Bunshin? Can a Kage Bunshin make another Kage Bunshin? 

Sometimes you see the clones die and I am screaming at the TV, why didn't that clone just make a clone of itself before it died?

Or does Naruto tend to let them die on purpose so he can gain their knowledge?
In short: Yes.

There are several instances where Naruto will create several Shadow Clones, which in turn create more Shadow Clones.

As far as why he doesn't create any more, this is speculative. I can think of a few reasons:


Naruto doesn't possess the skill to do so. The useful part of a clone creating a clone right before "death" would be to gain tactical advantage. Naruto's strategy is generally to "charge right in", while this kind of strategy would rely on the clones "dying" so as to be the bait for a new clone coming right in.
There simply isn't enough reaction time. Naruto has demonstrated that it takes time to create clones, even if the amount of time is relatively small. Rather than take the time to mold chakra and perform the technique, why not actually fight?