This talk is motivated by the following fundamental question: What is the logical/model-theoretic complexity generated by fractal objects? Here I will focus on fractal objects defined in first-order expansions of the ordered real additive group. The main problem I want to address here is: If such an expansion defines a fractal object, what can be said about its logical complexity in the sense of Shelah-style combinatorial tameness notions such as NIP and NTP2? The main results I will mention are joint work with Erik Walsberg.