If timestamp flag is set to 1, each router should add its outgoing address and the timestamp. This will take 42 bytes. Now theoretically an IP packet can be 65536 bytes long. A full header (that is all fields what so ever are used) will be 40bytes long. So we are left with ((65536-40)/42)=1559 routes (including IP address and time stamp). So theoretically answer is 1559 but in all practical purposes it is not the case as MTU never crosses 1500bytes and a route with 1559 will be never achieved because of TTL.
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