Before I start write a post about exFAT, it's good to know what it is:

exFAT (Extended File Allocation Table) is a proprietary file system designed especially for flash drives developed by Microsoft, which has applied for patent protection. It is supported in Windows XP and Windows Server 2003 with update KB955704, Windows Embedded CE 6.0, Windows Vista with Service Pack 1, Windows Server 2008, Windows 7, Windows Server 2008 R2 (except Windows Server 2008 Server Core), Mac OS X Snow Leopard starting from 10.6.5,[6] and Mac OS X Lion.
exFAT can be used where the NTFS file system is not a feasible solution, due to data structure overhead, or where the file size limit of the standard FAT32 file system (without FAT+ extension) is unacceptable.

via: Wikipedia
The exFAT file system is not supported natively on Linux because it is developed by Microsoft and it comes with restrictive license which doesn't allow open-source operating system to implement it natively.