The conventional wisdom is that chimpanzees have some level of self-awareness. For example, they can recognize themselves in a mirror. Dolphins and elephants may have similar abilities. From an AP story:
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.
This sounds more like an anecdote than a study, but you should probably stop eating elephant burgers, just in case.