It's well known that certain distributions have maximum entropy under moment constraints. For instance, maximum entropy H(X) under a variance constraint is given by the Gaussian distribution. Generally, for any positive integer k and for given constant c, there exist algorithms to find the distribution maximizing H(X) subject to E[X^k] = c.
However, here's what I learned yesterday: if we change the constraint to E[X^-k] = c, there exists no such maximum entropy distribution -- because you can always achieve unbounded entropy subject to the constraint.
At first I thought this was surprising, but then I realized it's not too hard to think of an example that shows why it's true. Think about it for a few minutes -- I'll post a hint in the comments.