The Fisher-Snedecor F distribution was recently proposed as an accurate and tractable composite fading model in the context of device-to-device communications. The present work derives the product of the Fisher-Snedecor F composite fading model, which is useful in characterizing fading effects in numerous realistic communication scenarios. To this end, novel analytic expressions are first derived for the probability density function, the cumulative distribution function and the moment of the product of N statistically independent, but not necessarily identically distributed, Fisher-Snedecor F random variables. Capitalizing on these expressions, we derive tractable closed-form expressions for channel quality estimation of the proposed model as well as the corresponding outage probability and average bit error probability for binary modulations. The offered results are corroborated by extensive Monte-Carlo simulation results, which verify the validity of the derived expressions. It is shown that the number of cascaded channels affects considerably the corresponding performance, as a variation of over an order of magnitude is observed across all signal-to-noise ratio regimes.