What happens to my LINK battery after 300 charge cycles?

The battery in LINK is guaranteed to operate at 100% capacity for 300 complete charge cycles, meaning that you can charge the battery from completely empty to completely full 300 times. If you charge your device every day but just a little bit to top it off, you will get far more than 300 charges.
Even after 300 charges the battery will continue to function normally, but the capacity will begin to diminish. At around 500 charges the battery life will be about 80% as long as it was when it was new. The LINK battery is not replaceable. 
In summary, you don't need to worry about your LINK dying on you! After 5 years you will need to start charging more frequently, but the battery should continue operating just fine.