Well, they were considered the animals nearest to man in intelligence

Elephants may be self-aware:

In a 2005 experiment, Happy faced her reflection in an 8-by-8-foot mirror and repeatedly used her trunk to touch an "X" painted above her eye. The elephant could not have seen the mark except in her reflection. Furthermore, Happy ignored a similar mark, made on the opposite side of her head in paint of an identical smell and texture, that was invisible unless seen under black light.

"It seems to verify for us she definitely recognized herself in the mirror," said Joshua Plotnik, one of the researchers behind the study. Details appear this week on the Web site of the Proceedings of the
National Academy of Sciences.

Still, two other zoo elephants, Maxine and Patty, failed to touch either the visible or invisible "X" marks on their heads in two runs of the experiment. But all three adult female elephants at the zoo behaved while in front of the jumbo mirror in ways that suggested they recognized themselves, said Plotnik, a graduate student at Emory University in Atlanta.

Maxine, for instance, used the tip of her trunk to probe the inside of her mouth while facing the mirror. She also used her trunk to slowly pull one ear toward the mirror, as if she were using the reflection to investigate herself. The researchers reported not seeing that type of behavior at any other time.

...

The three Bronx Zoo elephants did not display any social behavior in front of the mirror, suggesting that each recognized the reflected image as itself and not another elephant. Many other animals mistake their mirror reflections for other creatures.

That divergent species such as elephants and dolphins should share the ability to recognize themselves as distinct from others suggests the characteristic evolved independently, according to the study.

This recognition has so far only been recorded in monkeys, dolphins, and of course humans. But of course, more tests would need to be run to be sure.