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No seems kind of harsh. The player seems quite excited by the prospect of being incredibly old. I hate to squash creativity. Perhaps if he agrees to not be a magically active or potentially magically active type?
No offense meant with the "No", that's me speaking from bad experience with that kind of character.
Also,
all immortal elves are magically active, no exceptions. It comes with their dragion blood, I suppose. And they have access to a very, very powerful form of magic - though that seems to have been retconned out with newer installations of Frosty. Additionally, age should be reflected in power with magicans, and even with regula chatacters, thinkling of skills. Consequently built, this character would just be too powerful to handle, unless built into a specific niche.
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Looking through Harlequin, Harlequin's Back, and the Dawn of the Artifact stuff, it seems Frosty is an immortal elf. Powers: Immunity to Age, Disease, Pathogens, and Poisons per Harlequin's Back. Her initiate levels under H's tuteledge would give her the dragon magic access, not merely being immortal. Plus, she's not a spike baby.
So, perhaps a 25 Point Quality granting those 4 powers and some "antique" knowledge/interests/etc skills in the (Int+Log)*2 or 3?
Frosty is an Immortal and the daughter of Ehran the Scribe, Harlequin's arch-nemesis/buddy (depending on what book you read). Further, she's only some 45 years old (born in the late 2020s, I think?) and already a several-grades initiate that makes playing a mage in the Artifacts series nearly redundant.
Price it like Nosferatu, and build all characters according to high-powered campaign rules.
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The way I see it a spikebabie is just an elf with a long background story.
Well, yes. Spikes are regular elves that happened during a mana spike in the 5th world (conceived over the ruins of Sereatha or whatever). Spike babies are just unusually old regular elves (regular elves have a life expectancy of some 450 years). There may or may not be spike dwarfs, too.
Immortal elves are dragon/metahuman hybrids created by the dragons to create a race of gifted, powerful slaves to handle stuff that bores them while they are off doing dragon things (it didn't go well, and it didn't go particularly well with later experiments in that direction either). there may or may not be immortal dwarfs, too.
Your spike baby quality sounds okay, though I'd price it at 10 BP (20 Karma) for balancing reasons.