> a system can have high entropy but be relatively simple.
Good examples of these are anything that Kolmogorov-compresses well. For example, by almost any measure the output of a pseudo random number generator has high entropy. Yet it has low information density (low complexity), as the program that generates the sequence, plus its state, is really small.
I think a better example is just hot gas. Heat up a tube of gas, and its entropy will increase, with no effect on its complexity. Still not terribly compressible either though.