EDIT: this answer is wrong. I misread the question as looking at the group generated by P, not the points obtained by repeated doubling. I would be OK if the subset of S^1 generated by taking a non-torsion point and repeatedly doubling came arbitrarily close to the origin---but it may not, as the comments below show. As I write, this question is still open. If a correct answer appears I might well delete this one.
Original answer:
"Bounded" in what sense? You mention heights, that's why I ask. But in fact the answer is "no" in both cases. The height will get bigger because of standard arguments on heights. And the absolute values of x_n and y_n will also be unbounded: think topologically! The real points on the curve are S^1 or S^1 x Z/2Z and if the point isn't torsion then the subgroup it generates will be dense in the identity component and hence will contain points arbitrarily close to the identity, which, by continuity, translates to "arbitrarily large absolute value" in the affine model.
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