Elephants pass the "mirror test" for self-awareness:

In a series of experiments, the elephants first explored the mirror
— reaching behind it with their trunks, kneeling before it and even
trying to climb it — gathering clues that the mirror image was just
that, an image.
 

That was followed by an eerie sequence in which
the animals made slow, rhythmic movements while tracking their
reflections. Then, like teenagers, they got hooked.

All three
conducted oral self-exams. Maxine, a 35-year-old female, even used the
tip of her trunk to get a better look inside her mouth. She also used
her trunk to slowly pull her ear in front of the mirror so she could
examine it — "self-directed" behaviors the zookeepers had never seen
before.

Moreover, one elephant, Happy, 34, passed the most
difficult measure of self-recognition: the mark test. The researchers
painted a white X on her left cheek, visible only in the mirror. Later,
after moving in and out of view of the mirror, Happy stood directly
before the reflective surface and touched the tip of her trunk to the
mark repeatedly — an act that, among other insights, requires an
understanding that the mark is not on the mirror but on her body.

This is pretty extraordinary.  Besides bottle-nosed dolphins (who
also use mirrors to perform self-examinations), elephants are the only
non-apes to exhibit such behavior.