FAT, exFAT not readable!!

Have you installed third-party software such as "NTFS-3g" to make NTFS volumes writable in OS X? If so, either update it to latest version or remove it.

Any third-party software that doesn't install by drag-and-drop into the Applications folder, and uninstall by drag-and-drop to the Trash, is a system modification.

Whenever you remove system modifications, they must be removed completely, and the only way to do that is to use the uninstallation tool, if any, provided by the developers, or to follow their instructions. If the software has been incompletely removed, you may have to re-download or even reinstall it in order to finish the job.

I never install system modifications myself, and except as stated in this comment, I don't know how to uninstall them. You'll have to do your own research to find that information.

Here are some general guidelines to get you started. Suppose you want to remove something called “BrickMyMac” (a hypothetical example.) First, consult the product's Help menu, if there is one, for instructions. Finding none there, look on the developer's website, say www.brickmymac.com. (That may not be the actual name of the site; if necessary, search the Web for the product name.) If you don’t find anything on the website or in your search, contact the developer. While you're waiting for a response, download BrickMyMac.dmg and open it. There may be an application in there such as “Uninstall BrickMyMac.” If not, open “BrickMyMac.pkg” and look for an Uninstall button. The uninstaller might also be accessed by clicking the Customize button, if there is one.

Back up all data before making any changes.

You will generally have to restart the computer in order to complete an uninstallation. Until you do that, there may be no effect, or unpredictable effects.

If you can’t remove software in any other way, you’ll have to erase and install OS X. Never install any third-party software unless you're sure you know how to uninstall it; otherwise you may create problems that are very hard to solve.

Trying to remove complex system modifications by hunting for files by name often will not work and may make the problem worse. The same goes for "utilities" such as "AppCleaner" and the like that purport to remove software.

sudo fsck_exfat /dev/disk2s1

The above is an example of the disk check command for the Terminal. You should run this for exFAT volumes. They are not journaled like HFS, but instead like FAT, with an allocation table, file table and boot record. These can be repaired by OSX, but not always from disk utility; it sees the files but they are corrupted, so it assumes msdosFAT32 as a format, and attempts to repair them as such. They are not FAT32 compatible. They are instead an extended form; hence the EX in exFAT, meaning extended. While there are several versions of exFAT, all of them utilize the standard allocation set of records. Unfortunately, these formats are often slow unless on newer interfaces like thunderbolt or usb3, and corruption can occur easily. I recommend you do the following:

open disk utility, select the partition with the problem (if you can see it; if not, we'll get to that);

next, glick on the "info" icon, a blue "i" button above;

Look for the entry similar to the one above, the mount point or disk enumerator--looks like /dev/disk#s#;

now open terminal and type the following command:

sudo fsck_exfat /dev/disk#s#

be sure to replace the "/dev/disk#s#" with the information you got from the info window. Now just sit back and watch the fireworks. When it finishes the scan, it may ask you to repair, just type "y" and then "return\enter".

I cannot guarantee this will work if your disk has too much hardware damage, but in many instances it has saved me from a reformat.