What is the shortest description of an object? The size of this description, is the descriptional complexity. This general notion may also be called "structural complexity".
See also Complexity theory, for other notions of complexity.
Lecture notes on descriptional complexity and randomness
Based on the minimum size of a program (interpreted by a Turing machine) that produces (describes) the object.
Effective complexity, and its related to an Algorithmic minimal sufficient statistic, I think are best candidates towards a measure of complexity which agrees with our intuition :) – Logical depth (paper) seems related. Complex models require more computation/thinking (c.f. adaptive computation time..),
Time-bounded Kolmogorov complexity can give a way of looking at Causality, related to that of logical depth.
YB videos: https://www.youtube.com/watch?v=HWsa_hZ7F3I