The success chance will be a lot lower if the gem is a higher rank than the greater rift level is. The upgrade chance for success is based on a combination of the greater rift level you're on and the gem's current rank. Upgrading a gem does not always work, though. You can use all 3 attempts on the same gem or on different ones. Upon completing the greater rift, you will receive 3 chances to upgrade or “rank up” your gem. You can have the chance to upgrade a legendary gem every time you complete a greater rift. At rank 25, an additional effect is unlocked. When first acquired, legendary gems start at rank 0, and can be upgraded to become stronger. When choosing to upgrade legendary gems, you will end your current greater rift run as you will be unable to get a new keystone from Urshi at the same time. Legendary gems can be upgraded at Urshi, a nephalem ghost NPC which spawns after completing a greater rift. Only gems your character does not yet own (equipped, in the inventory, or the stash) are being dropped, so getting all of them will not take long. Gem drop chance is very high, with an average between 1 and 2 drops per kill.

The 14th one is Boon of the Hoarder, which comes from Baroness Greed in The Vault (goblin realm). 13 of them are acquired from greater rift guardians. Legendary gems can only be socketed into rings and amulets. I've found a nice compilation of information on Legendary Gems on reddit, made by Aesir81, source here:

Diablo 3 upgrade legendary gems