What productive tasks can you achieve when forced to sit through a boring meeting? Things that can be done in an environment with continual audible distractions and occasional interruptions needing your attention.
Here is one strategy from pp. 51-2 of Csikzentmihalyi's book on Flow: "Heinz Maier-Leibnitz, the famous German experimental physicist ... provides an intriguing example of how one can take control of a boring situation and turn it into a mildly enjoyable one. ... To alleviate [the burden of sitting through boring conferences] he invented a private activity that provides just enough challenges for him not to be completely bored during a dull lecture, but is so automated that it leaves enough attention free so that if something interesting is being said, it will register in his awareness. ... Whenever a speaker begins to get tedious, he starts to tap [his fingers in a regular pattern] ... there are 888 combinations one can move through without repeating the same pattern... Professor Maier-Leibnitz found an interesting use for it: as a way of measuring the length of trains of thought...Suppose a thought ... occurs in his consciousness while he is tapping during a boring lecture. He immediately shifts attention to his fingers, and registers the fact that he is at the 300th tap of the second series; then in the same split second he returns to the train of thought. [When the train of thought is complete, he calculates how far the series of tapping progressed while he was thinking.]"
The reason that so many useless meetings exist is that no one has to float the bill. The second some suit in a different department needs to justify the expenses of calling too many people to too many non-productive meetings they diminish drastically.