The paper is concerned with observation of discrete-time, nonlinear, deterministic, and maybe chaotic systems via communication channels with finite data rates, with a focus on minimum data-rates needed for various types of observability. With the objective of developing tractable techniques to estimate these rates, the paper discloses benefits from regard to the operational structure of the system in the case where the system is representable as a feedback interconnection of two subsystems with inputs and outputs. To this end, a novel estimation method is elaborated, which is alike in flavor to the celebrated small gain theorem on input-to-output stability. The utility of this approach is demonstrated for general nonlinear time-delay systems by rigorously justifying an experimentally discovered phenomenon: Their topological entropy stays bounded as the delay grows without limits. This is extended on the studied observability rates and appended by constructive finite upper bounds independent of the delay. It is shown that these bounds are asymptotically tight for a time-delay analog of the bouncing ball dynamics.