In an IP packet there is a Time to Live(TTL) 16 bit number that is used to *set* the max hop count. The original host sets this TTL field and then each successive Layer 3 devices the packet passes through decrements the TTL and then checks if it is greater than 0. If it is not greater than 0 then it will discard the packet and send an ICMP message to the original sender of the packet saying that the packet was discarded because the TTL field got to 0. The max a TTL field can be is 2^16 = 256 or 255(since it starts at 0, not 1).
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