Octopuses have the largest brains of any invertebrate. Athena's is the size of a walnut -- as big as the brain of the famous African gray parrot, Alex, who learned to use more than one hundred spoken words meaningfully. That's proportionally bigger than the brains of most of the largest dinosaurs.
Another measure of intelligence: you can count neurons. The common octopus has about 130 million of them in its brain. A human has 100 billion. But this is where things get weird. Three-fifths of an octopus's neurons are not in the brain; they're in its arms.
"It is as if each arm has a mind of its own," says Peter Godfrey-Smith, a diver, professor of philosophy at the Graduate Center of the City University of New York, and an admirer of octopuses. For example, researchers who cut off an octopus's arm (which the octopus can regrow) discovered that not only does the arm crawl away on its own, but if the arm meets a food item, it seizes it -- and tries to pass it to where the mouth would be if the arm were still connected to its body.
"Meeting an octopus," writes Godfrey-Smith, "is like meeting an intelligent alien."
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Octopuses consist almost entirely of hydrostatic limbs and probably there are more sensory nerve fibres innervating the limbs as well. This means the octopus can have a disproportionate brain-body mass ratio compared to typical animals without being super intelligent in the sense of being able to write any Broadway musicals.