Kierkegaard tells us that the more we are aware of self, the more self we have. For him, this is a positive outcome since being an aware self is the prerequisite for a relationship with that "power which constitutes it", its Creator. The alternative would be to be bound within an encapsulated self which cannot see beyond itself. As alien as this may seem to those of us of a more "Eastern" persuasion where self is generally seen in a negative light, still I think Zhuangzi might have smiled in agreement with such a view.
To be self-aware is to be a self that sees itself. What does it see? It sees its foibles, its stupidities, its smallness. But if it sees itself, then it more than itself; or rather, it is more than its encapsulated self. Zhuangzi's hypothetical sage who has "no self" might also be described as one who has so much self that she doesn't have to be that ridiculously encapsulated self. She also "has no name", but my guess is that if we called her name she would look up.
One manifestation of being self-aware is the ability to laugh at one's self. Zhuangzi does this all the time. He makes pronouncements, then admits that they are groundless. There is good evidence that the whole of the Inner Chapters are a kind of joke. Zhuangzi figures that if he can get us laughing we might be set free from the seriousness that accrues to fixation on . . . well, things "fixed". It's hard to imagine Xunzi with all his rights and wrongs and pseudo-rational proofs of the truth of his way having a good laugh at himself. No, this is all very serious stuff, this spirituality. This is perhaps why he is able to so readily endorse the "five punishments" (tattooing, nose chopping, foot lopping, castration, and death).
One reason I love the humor of Woody Allen is its self-deprecating nature, as for instance, when he admits to being one of the 'few' males who experiences "penis-envy", an attribute that Freud intended only as descriptive of females. (A projection, no doubt.) This kind of self-awareness obviously does not make one a sage, but it's a good start.
To be self-aware is to have a self to play with. Monopoly is fun because it's a game. When we know it's a game we have fun; when we take it seriously we get angry, or hurt, or vengeful, or throw a tantrum. He who can play with his self, is also more likely to able to play well with others.
To have this larger self to which Kierkegaard aspired is to experience a kind of self-transcendence. This may not meet the specifications of no-self and other more radical pronouncements of ultimate spirituality, but for us novices in the way, it's a great first step.
You can check out Scott's writings on Zhuangzi here.