Ramana Maharshi claimed you could unveil your true nature by constantly dwelling on the question, "Who Am I?" The practice of self-inquiry has been a doorway to insight for eons. The more we look, according to both science and Buddhism, the more we into the nature of things.
One of the cornerstones in Buddhist practice, which many say is the single-most profound contribution of the Buddha, is the concept of anatta, or 'no-self.'
This is a gnarly thing to try to wrap our brains around because to fully grasp it, we have to go beyond the linear, rationale mind. As I understand it, to dwell on the philosophy of 'no self' is pretty much a waste of time and was actively discouraged by the Buddha. The practice, though, can lead to profound insights into the nature of self.
The following clip is from a program exploring the illusion of the human self. The first portion explores a bit about memory and self. At about 4:33, Eckhardt Tolle speaks on the topic in his interview with Oprah last year. I find his western perspective quite refreshing.
If you're interested, I'll be leading a daylong retreat of inquiry into the question "What Am I?" in May.