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Does anyone know what the theoretical maximum number of skills it'd be possible to gain in Dead Reign?

Well that depends on what the characters have access to. And if you are using rules options from other books. In Heroes Unlimited 2nd Edition there is a blurb that covers characters learning skills by going to school like how we learn stuff in real life. That rule set is not exactly restricted to HU2 it could apply to any setting that uses education and skills (basically all of them). With minor modification it could be adjusted to cover using "do it yourself", "how to" and similar educational books to teach yourself skills. Given that, by my count there looks to be around 97 skills in the book that could be book or trial and error learned, technically if the character lived long enough or had a safe haven with a fully stocked library where they weren't having to fight to survive day by day they could potentially learn all of those skills, more if you take into account learning additional languages.

The only maximum number of skills you can have is based upon the skills in the book. You can conceivably have all the skills in the book with the proper application of brain power. They would count as secondary skills though. But that's just me.