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Gonna talk about the battles in spoilers just in case.
Gilgamesh battle is utterly ridiculous. From watching my roommate play the battle, the first bit seems simple enough. You beat him super easy with those silly guns of his.
Then he pulls out his swords 'cause he's like "Lolno" and he gets terrible. All damage, elemental or not, is halved and he's immune to wound damage. His HP sits at 9,999,999 if I read it right. Which I'm pretty sure I did. That's right, pretty much ten MILLION HP. Near as I can tell you have to stagger him, at which point he becomes weak to all elements and no longer immune to wound damage. He uses an ability called Eerie Soundwave which I think inflicts random debuffs while inflicting damage. When his stagger runs out...
He heals to full HP. Yep, that's right. My roommate dealt over 1 million HP worth of damage to him and he just "BAM" got it back at the end. If I'm right, you have to stagger him, wound the shit out of him, and repeat. My roommate gave up though, because Gilgamesh was dealing all kinds of wound damage and he didn't have any wound potions. It's evil.
Then there's the PuPu fight. You're fighting six PuPus AND their UFO mothership. The PuPus themselves seem relatively weak, and you can beat them pretty quick. But they can't be staggered, and the UFO will use potions to heal them. The UFO will also sometimes gear up to use a super attack which I THINK may be like a Thundaga laser from the sky. But honestly, there's too much going on in this screen for me to tell. It does seem to damage based on the number of PuPus on the screen, so as you can imagine you should take them out as quickly as possible. Especially since the UFO drops off more PuPus sometimes, and I think it might be based on time as well as number of PuPus remaining on the field. When you stagger the UFO though (the only thing on the screen with a stagger gauge) all the PuPus will run around in circles bumping into each other and generally doing nothing.
My roommate didn't finish this fight either, but it was because his only defensive paradigms were for healing/buffs (Sen-Med-Med and Sen-Syn-Syn) and he couldn't keep the stagger gauge filled. I think your chance to get a PuPu might be with every one you kill, but I have no way of testing it at the moment.
Yeah, getting PuPU's crystal is very easy despite the lower drop rate. I only had to battle it twice before I was able to get it, on account of you killing like 12~18 per battle.