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We consider an ultra-dense wireless network with N channels and M = N devices. Messages with fresh information are generated at each device according to a random process and need to be transmitted to an access point. The value of a message decreases as it ages, so each device searches for an idle channel to transmit the message as soon as it can. However, each channel probing is associated with a fixed cost (energy), so a device needs to adapt its probing rate based on the "age" of the message. At each device, the design of the optimal probing strategy can be formulated as an infinite horizon Markov Decision Process (MDP) where the devices compete with each other to find idle channels. While it is natural to view the system as a Bayesian game, it is often intractable to analyze such a system. Thus, we use the Mean Field Game (MFG) approach to analyze the system in a large-system regime, where the number of devices is very large, to understand the structure of the problem and to find efficient probing strategies. We present an analysis based on the MFG perspective. We begin by characterizing the space of valid policies and use this to show the existence of a Mean Field Nash Equilibrium (MFNE) in a constrained set for any general increasing cost functions with diminishing rewards. Further we provide an algorithm for computing the equilibrium for any given device, and the corresponding age-dependent channel probing policy.