Whelk

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Whelk
HP: 1,600, MP: 1,000, Weaknesses: None, Nullifies: None, Absorbs:Thunder
 
TekMissile go!

Being the first boss of the game, it's quite easy. Just attack the head with ?????'s TekMissile attack and Vick's and Wedge's Fire Beam, and it should go down quickly. Also, follow Vicks' and Wedge's advice about not attacking the shell. If you attack it will counter with Mega Volt, which will do around 60 damage to one character. Don't queue up too many attacks, or you will hit the shell when the Whelk's head retreats into its shell.

You COULD try taking out the shell. It is easier said than done, as the shell can only use Mega Volt six times before it runs out of MP. Now, it's only the problem of taking out the shell's 50000(!!) HP before you die of boredom... Your reward is a Tincture instead of Potion, and the feeling you have just done one of the most pointless tasks you will ever encounter in your life...

Marshal

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Marshal
HP: 420, MP: 150, Weaknesses: Poison, Nullifies: None, Absorbs: None

Lobo x2: HP: 27, MP: 5, Weaknesses: Fire, Nullifies: None, Absorbs:None

 
Mog's Dance skill can be quite useful in this battle
This is a fairly straight forward battle. Just kill the Lobos first, then concentrate on the Marshal. Heal when necessary. If you've got Mog in the party, have him Dance Dusk Requiem. He'll fall quick enough.

Vargas

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Vargas
HP: 11600, MP: 220, Weaknesses: Poison, Nullifies: None, Absorbs: None
This is actually a fairly trivial battle to win. When Edgar, Terra, and Locke face Vargas have them attack the bears first and then finally attack Vargas himself. Terra should use magic, Edgar should use the autocrossbow. After the fight has gone on for a while, he'll blow the party away and Sabin will show up to help them. All Sabin needs to do to win the fight is to perform the pummel blitz by selecting blitz, hitting left, right, left, then hitting A. Surprisingly, a lot of people gives up this game at this point as they don't know how to Blitz.

Ultros: First Battle

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Ultros
HP: 3000, MP: 640, Weaknesses: Lightning and Fire, Nullifies: None, Absorbs: None
Get used to fighting this guy, you'll be seeing a lot of him. Everyone's moves are pretty simple. Terra should continuously cast Fire for massive damage, Sabin should use Fire Dance (or Aura Bolt if you don't have Fire Dance, which is learnt at level 15), Edgar should use the autocrossbow, and Banon should cast Health every time. Switch characters by pressing X if you don't need to restore your health and want to save Banon's turn for later.

Tunnel Armor

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Tunnel Armor
HP: 1300, MP: 900, Weaknesses: Lightning and Water, Nullifies: None, Absorbs: None
Have Celes use Runic every turn to absorb its magic attacks and have Locke attack. If you're hurt (by his physicals), have Locke heal with potion and such. You can also steal an Air Lancet from him.

Ghost Train

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Ghost Train
HP: 1300, MP: 900, Weaknesses: , Lightning, Pearl, Cure, Nullifies: None, Absorbs: None
This can either be a challenging boss fight or a laughably easy one. Since the boss is undead cure magic and items will diminish its HP, and using a Fenix Down will kill it instantly. If you want to fight it the fair way, just use aurabolt, dispatch, and have Shadow throw Shyruiken. If you want a laugh, use Suplex.