Welcome to the boss section.  This covers only the Sanctuary bosses and, of course, Gigyas...
First Sanctuary: Titanic Ant
HP: 235, Off: 19, Def: 23

Strategy: Basically, I've found you have a one in 164 chance that you will have a bomb from a rowdy mouse in the cave.  So, if you get one, great, but don't try for one.  OK, easiest way to defeat Titanic Ant is to use PSI Rockin' once to knock out his two buddies, and then bash him, and heal when nessecary.  Remember if you can only stand one more attack, heal because he's faster than you.  Eventually, you should defeat him.  If you get a bomb, use it after knocking out his two buddies because it does somewhere around 100 with PSI doing around 80, so he would be almost defeated!
Note: PSI Rockin' is refering to Ness's favorite thing and is his only damaging offensive attack
Second Sanctuary: Mondo Mole
HP:  498, Off: 37, Def: 50

Strategy:  This has to be the easiest or the hardest boss ever, depending which way you go.  The hard way is to just start bashing away.  Paula will go down with one hit, abut you will eventually win if you keep healing Ness.  The easy way is to use Paralysis alpha, and Mondo Mole will be paralyzed!  Then Paula will get that experience and level way up!
Third Sanctuary: Trillionage Sprout
HP: 1048, Off: 54, Def: 88

Strategy:  As said up top, You don't have to go in order!  So I save this guy 'til last and he's a pushover!  Using PSI Fire Omega really puts the whop on him!  Plus Poo is with you, so Trillionage sprout is so easy!
If you prefer to take the hard way, use PSI Fire Beta until his two toadies are gone, and then just keep bashing.  Hopefully you have some horn of life's, because he does glare with his eerie eyes which diamondizes your person if it works!
Note: You actually don't have to go in order with the Sanctuary Bosses!
The only one's you have to fight to continue the story are the 1st, 4th, and 7th, but I suggest fighting the second to help Paula lvl up, and you cannot fight the 6th without fighting the 5th first and getting the carrot key.
All Strategy's here are the one's I use, and you can use your own strategy if you like, and you can use Bombs and Bottle Rockets instead of Bashing if you like, even if you follow my strategy.
Fourth Sanctuary: Shrooom!!
HP:1700, Off: 95, Def: 157

Strategy:  You have to fight this guy to continue your story, so he's not such a pushover.  Before you fix the Sky Runner in Threed, go to Twoson and buy at least 5 teddy bears.  When you fight Shrooom!, he will constantly try to do that mushroom thing on the teddy bear, which won't affect him!  Just keep bashing and using Fire Beta or Gamma and you should defeat him.  If you go into battle without teddy bears, use the strategy above, but instead, if he makes a mushroom on one of your people, have them defend.  If he poisons one person, have Ness heal them unless he's mushroomized.  If all your people are mushroomized, have Ness and Jeff defend and have Paula pray.  The second strategy takes a super long time, but at least you will defeat him.
Fifth Sanctuary: Plague Rat of Doom
HP: 1827, Off: 74, Def: 180

Strategy:  You can fight this guy after you beat the 8th sanctuary boss, but you can't fight the 6th without the carrot key in the sanctuary area.  If you fight this guy after the 8th, he will be somewhat of a pushover.  He still gives you that problem that he has high guts, which makes him SMAAAAAAASH!! often.  Just bash him and heal as necessary.  If you battle the guys in order, this guy will be tough to get to without having Poo die.  Then battling him without having Poo die will be even tougher.  (Hint: Train Poo out in the desert between Threed and Fourside.  Then Plague Rat of Doom won't be so tough)
Sixth Sanctuary: Thunder and Storm
HP: 2065, Off: 111, Def: 171

Strategy:  This guy is an absolute pushover if you wait until after the 8th sanctuary boss, because Ness should know PSI Flash Omega.  If you use that, Thunder and Storm should just die right away!!  If you fight them in order, have Ness use the strongest flash he has.  Then just keep bashing away.  If Thunder and Storm become paralized, then you don't have to worry much, just about Thunder Beta and Flash Beta, which is disguised as Crashing Boom Bang and creating a storm.  Make sure Ness has the Night Pendant and the Franklin Badge, and at least Ness will make it through the battle.
Seventh Sanctuary: Electro Spector
HP: 3092, Off: 148, Def: 203

Strategy:  This guy is kinda tough.  Just keep bashing away and healing as necessary.  The only thing you have to worry about is his Thunder Beta attack.
Eighth Sanctuary: Carbon Dog/ Diamond Dog
Carbon Dog: HP: 1672, Off: 152, Def: 174
Diamond Dog: HP: 3344, Off: 167, Def: 230

Strategy:  I this guy is hard in both forms.  First, when facing Carbon Dog, have Paula use Freeze beta, and have the rest bash, but be sure that Paula has the Flame Pendent and Jeff has the Sea Pendent.  This guy hurts your whole team with his fireball attack, and if Paula and Jeff are equiped like above, then this should do only like 4 - 12.  When he has gotten enough damage, he changes into Diamond Dog.  If you used a physical attack to finish Carbon dog, your attack will be bounced back.  Diamond Dog has Sheild Beta, which rebounds your attack.  Just continue to bash and heal as necessary.  (Note:  I have been told that Diamond Dog can diamondize you with his bite, but I have never had it happen to me, so I don't know)
And now, the final battle with Giygas
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