What animal has the most hearts?

The octopus

An octopus has three hearts. Two hearts are associated with the gills and are called gill hearts.

https://www.newscientist.com/question/many-hearts-octopus

As I vaguely recalled from high school biology, the earthworm has five hearts.

http://extension.illinois.edu/worms/anatomy/anatomy6.html

However, I think much happens on what you actually call a heart.

A typical lumbricid (earth worm) will have 5 hearts.

https://en.wikipedia.org/wiki/Earthworm

The cockroach has (arguably) 13 hearts.

https://www.medgadget.com/2009/06/artificial_heart_thats_inspired_by_a_cockroach_and_built_like_an_onion.html

The earthworm has five pairs of hearts, which would make a total of ten. Dissection (the reason I avoided Biology in High School!) instructions:

"Follow the dorsal vessel anteriorly to the esophagus and find the five pairs of large, contractile, segmental blood vessels which are the hearts ."

(lander.edu, no longer online)

Some describe the cockroach as having 13 hearts, others describe it as having a 13-chambered single heart. "A cockroach has a chain of hearts, in all totaling to 13. Some regard it as a single heart with 13 chambers. These are funnel shaped chambers with a pair of tiny holes called ostia."

(suite101, no longer online)

As was said before, this depends on how you define heart. There is such a thing as a pseudoheart, and earthworms are said to have pseudohearts. Similarly, a cockroach has a "a single structure with 12 to 13 chambers."

https://en.wiktionary.org/wiki/pseudoheart

Cephalopods (cuttlefish, squid, octopus) have three hearts: two for the gills and one central heart, and that is the correct answer.