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She's trying to read as many books as possible.
Literal
She [topic-は] [as much as possible-できるだけ] many [genitive-の] books [object-を] [is-trying-to-read-読もうとしている].
Builds three modifiers in front of 本: できるだけ ('as many as possible') + たくさん ('many') + の ('attributive') = 'as many books as she can.' The volitional 読もうとしている ('is trying to read') describes ongoing effort — the volitional form ~(よ)う + とする marks an attempt, and ている freezes it as a sustained state. Together: someone in the middle of an active reading project, not just settling down with a single book.