Genetic information often exhibits hierarchical and nested relationships, achieved through the reuse of repetitive subsequences such as duplicons and transposable elements, a concept termed ``evolutionary tinkering'' by Fran\c{c}ois Jacob. Current bioinformatics tools often struggle to capture these, particularly the nested, relationships. To address this, we utilized Ladderpath, a new approach within the broader category of Algorithmic Information Theory, introducing two key measures: order-rate $\eta$ for characterizing sequence pattern repetitions and regularities, and ladderpath-complexity $\kappa$ for assessing hierarchical and nested richness. Our analysis of amino acid sequences revealed that humans have more sequences with higher $\kappa$ values, and proteins with many intrinsically disordered regions exhibit increased $\eta$ values. Additionally, it was found that extremely long sequences with low $\eta$ are rare. We hypothesize that this arises from varied duplication and mutation frequencies across different evolutionary stages, which in turn suggests a zigzag pattern for the evolution of protein complexity. This is supported by simulations and studies of protein families like Ubiquitin and NBPF, implying species-specific or environment-influenced protein elongation strategies. The Ladderpath approach offers a quantitative lens to understand evolutionary tinkering and reuse, shedding light on the generative aspects of biological structures.