I would like to build a box counter for machine, so I'm going to use EEPROM in Arduino to save count. But the lifecycle is limited to 100,000 writes. Please suggest an external EEPROM with a long lifecycle or any other solution.
You can cycle through all the addresses in the EEPROM like a ring buffer.
You'll need to reserve a bit so you can binary search for the head of the buffer during startup.
You can also reduce how often you write in general especially when the count will increment in bursts, try and wait out those bursts instead of writing on every increment.
But for a device which writes a lot, a better solution would be a NVRAM, FRAM or EERAM. Those usually come with virtually no limitations to the write cycles but usually are more expensive than EEPROMs but that might very well be worth it compared to the hassle you can get in otherwise.
If you use something like an SD card, they have internal logic that will move the data around when a call is used up. This should give you virtually unlimited storage for something as simple as a count.