Animals with no Ears
While they can't quite be said to have no ears at all, a difference of a single gene can lead to cats with much smaller droopy ears.
This mutation is the defining feature of cat breeds like the American Curl and Scottish Fold (pictured).
Cats are also sometimes born entirely without external ears, e.g. Emma (2008), Frazzle (undated). This trait is linked to the Persian breed and long-haired cats.
Other cats will lose their ears due to frostbite or removal of cancerous tissue (as with the cat shown below).
Photo credit: MICHAEL QUICK of "MICHAELS PHOTOGRAPHY de NEMOURS" / Foter.com / CC BY-NC-ND
Weasel the terrier(2007) was born not only without pinna (external ear-flaps), but lacking even an entrance to the auditory canal.
Phoenix (2010) is an earless German shepherd. His ears were cut of as part of his use as a bait dog for training fighting dogs. But his story ended well when he was rescued and adopted.
Puppies without ear flaps and sometimes with closed ear canals seem to occur more often in hairless breeds.
The first animal I ever came across that was genetically/naturally earless was a goat at a county fair. The breed originates from Spain and as referred to as a "La Mancha". They are now popular throughout the United States.
They have very small, nub-like ears and come in many different coat colors and patterns.
Creole the hamster (2011) was found in a pet store amongst normally eared hamsters. Apparently this issue is not uncommon in hamsters bred for pet stores.
This unnamed horse (2011) is deaf apparently due to a congenital defect that include loss of external ears.
Wild pigs without ears have been found by hunters at several different locations.
Rabbits are born without ears fairly regularly. A case in Japan recently was linked to a fear of the effects of radiation.
The ear-less rabbit is, however, most likely just a spontaneous occurrence as with other known cases like Xavier (Singapore, 2006), Vincent (United Kingdom, 2008), two unnamed rabbits (China, 2011) and Til (2012). Til was literally a victim of his own fame, dying by being stepped on by a camera man.
Other rabbits lose their ears of maternal over-grooming, cold or abuse.
See also: Vintage photograph of rabbit with no ears.