It seems like every anime has one, although it's much more common in comedy and shoujo anime. Sometimes they exist only to pop up on screen every third episode. A small, painfully adorable creature, the obligatory cute anime animal is usually some sort of mythical creature, so it can be even more kawaii than any actually existing animal. Failing that, it's often a cat.
Kero-chan (Card Captor Sakura)
Jiji (Kiki's Delivery Service)
Chuchu (Shoujo Kakumei Utena)
Mokona (Magic Knights Rayearth)
Tama (Fushigi Yuugi)
That little black cat (Trigun)
Ryo-Ohki (Tenchi Muyo)
The welsh corgi (Cowboy Bebop)
All those cats (Sailor Moon)
do I even need to mention Pokemon?
And that's just off the top of my head…