 If you liked
Damekko Doubutsu
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 ...then you might like
Chi's Sweet Home
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Primarily, both involve animals and have short episode lengths - Chi's Sweet Home with episodes around 2:30 minutes each and Damekko Doubutsu with times around 5:00 minutes in length. They're both sweet and comedic, and both are meant for children. Damekko doesn't have the 'Awww' factor that Chi does, however, but they are both similar in other respects.
They are both pure cuteness with a bit of humor and the episodes are extremely short (although Chi's Sweet Home are much shorter). Both have to do with talking animals and are pretty random with no real plot. Both are great to watch in one long boring day.
They both have an animal theme. They both have adorable characters. Episodes that last 5 minutes or less. No plot line but still enjoyable.