Famfrit is fought during a regular playthrough of the game when the team reaches the top of the Pharos at Ridorana and battles Dr. Cid for the final time. He is automatically obtained after the end of the battle.
Famfrit is fought in the top room of the Pharos at Ridorana which is called the Womb of the Sun-Cryst.
The description below includes the full battle which also involves fighting Dr. Cid.
Level: 50 and 52
HP: 82,093 and 149,060
Steal: 
Hi-Potion (55%), 
Ketu Board (10%), 
Magepower Shishak (3%) and Aquarius Gem (55%), 
Elixir (10%), High Arcana (3%)
This boss battle is broken out into two distinctive sections – before and after Famfrit joins the battle.
For the first part of this battle you will face just Doctor Cid alone. Once he reaches 50% HP he will summon Famfrit to join him in the fight.
The first battle against Doctor Cid is extremely straight forward. Cid will only use regular physical attacks. Prepare ahead for the more difficult upcoming battle by using this time to renew all of your beneficial status effects such as 
Protect, Shell, 
Bubble and 
Haste. You may want to consider equipping your healer with a 
Rose Corsage in preparation for the next battle.
At 50% HP, Doctor Cid will use his manufactured Nethicite summon Famfrit to do battle by his side. Doctor Cid will become immune to attacks while Famfrit is summoned. Switch your Attack Gambits to attack the Foe with the highest level and this will cause your party members to properly shift to Famfrit when it is summoned and back to Doctor Cid as soon as it is defeated.
Remove any Dispel Gambits that you have setup for the moment as your party members will attempt to Dispel Doctor Cid which will not work until Famfrit is defeated. As a reminder, your healer should have a Rose Corsage equipped and should be setup with an Esuna Gambit to remove the effect from other party members. Alternatively, setup a Gambit to use 
Echo Herbs on party members who are Silenced.
Both Famfrit and Doctor Cid are weak to Fire-based Magick which makes using Oil on either of them extremely successful. Famfrit will increase its defense when it reaches low HP and it will use some special attacks during the fight that cut into a cinematic.
Go back to attacking Doctor Cid once Famfrit has been defeated. Doctor Cid’s attacks will now do more damage.
Congratulations – you now have Famfrit as an Esper which you can use in battle!
Famfrit has two Water-type attacks. The first, his standard attack, is Briny Cannonade, which deals water damage to one party member. His ultimate attack, Tsunami, causes a maelstrom that hits all foes in range with waves of water.
Unlocking Famfrit on the License Boards of certain jobs will unlock additional License Boards:
Vanishga, Warp), Time Magick 9 (Reflectga, Slowga), Time Magick 10 (Graviga, Hastega)Dispelga, Arise)
Numerology, Daggers 5 (
Orichalcum Dirk, Platinum Dagger)The optimal selection, assuming you are using a Machinist, is to give them all of the Time Magick spells available. Alternatively, you can stack your Archer or Black Mage with HP Licenses.
