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We propose a new five-parameter entropy function that proves to be singular-free during the entire cosmic evolution of the universe, and at the same time, also generalizes the Tsallis, Barrow, Rényi, Sharma-Mittal, Kaniadakis and Loop Quantum Gravity entropies for suitable limits of the parameters. In particular, all the above mentioned known entropies become singular (or diverge) when the Hubble parameter vanishes in course of the universe's evolution (for instance, in bounce cosmology at the instant of bounce), while the newly proposed entropy function seems to be singular-free even at $H = 0$ (where $H$ represents the Hubble parameter of the universe). Such non-singular behaviour of the entropy function becomes useful in describing bouncing scenario, in which case, the universe undergoes through $H = 0$ at the instant of bounce. It turns out that the entropic cosmology corresponding to the singular-free generalized entropy naturally allows symmetric bounce scenarios, such as -- exponential bounce and quasi-matter bounce scenario respectively. In the case of exponential bounce, the perturbation modes are in the super-Hubble domain at the distant past, while for the quasi-matter bounce, the perturbation modes generate in the deep sub-Hubble regime far before the bounce and hence resolves the ``horizon issue''. Based on this fact, we perform a detailed perturbation analysis for the quasi-matter bounce in the present context of singular-free entropic cosmology. Furthermore the entropic cosmology in the present context is shown to be equivalent with the generalized holographic cosmology where the holographic cut-offs are determined in terms of either future horizon and its derivative or the particle horizon and its derivative.