How do you increase your pet's treasure chest? I know sometimes the pet zappers will increase them, but isn't there another way?

Zappers, random events, and Keepsake boxes can increase your pets' treasure chests. :)

The Magical Keepsake Box can be found at the Millionaire Center and the Multi-Compartment Keepsake Box can be found in the Cash Shop.

Magical Keepsake box -in the MC for 150k, 1-3 spaces.

And the Multi-compartment box in the Cash shop, 10-15 spaces.