In this article, we lay solid foundations for the study of Maximal Entropy Random Walks (MERWs) on infinite graphs. We propose an appropriate definition that extends the original one from finite to infinite setting. Unlike conventional simple random walks, which maximize entropy locally, MERWs maximize entropy globally along their paths, representing a significant paradigm shift. They were introduced by physicists and computer scientists in \cite{ref19} and are characterized by strong localization properties in heterogeneous environments. In the infinite setting, however, MERWs are not always unique and might not even exist. Despite this, they continue to maximize the entropy rate, albeit in a less direct way. Our investigation focuses on weighted spider networks, where we explicitly describe the MERWs and highlight a phase transition phenomenon related to the model's parameters. In studying the scaling limits of these random walks, we obtain some interesting Walsh diffusions and the three-dimensional Bessel process. A unified proof is presented based on submartingale problems. Notably, when applying our findings to particles satisfying to the Pauli exclusion principle, we can deduce the electrostatic force by solely relying on the maximal entropy principle. We also touch upon how to achieve these results for continuous time stochastic processes, utilizing the Kullback–Leibler divergence.