For an algebraic number $\alpha$ one can define its "height" in many ways. Informally, you could use its minimal polynomial over $\mathbf{Q}$ and consider the maximum of the heights of its coefficients. Or consider all the valuations of $\alpha$, etc. In this context, the height is supposed to be some kind of measure of complexity.

**Question.** Is there a reasonable definition of the "height" of a transcendental number.

I'm not sure what such a height would mean though in this context.

If there isn't any reasonable definition, is there any reasonable explanation for why this isn't possible?