As said before, Paragon ntfs is an excelent program that let you read and write ntfs-disks on a mac.
And there is a similar program to read mac-disks on a windows computer, and that is Macdrive.http://www.mediafour.com/products/macdrive/downloads
But there is nothing wrong with exfat, I am using it most of the time when transfering data between mac and windows. (and linux)
The reason windows screws up repairing the exfatdisk is the different way the rubbish-bin is inplemented.
let windows repair the disk, tell it to shut up and get to work, and it will do its job.
Fat-32 can be used too, almost any computer will read and write it.