It should be hurt. Anyone takes a singular verb when it is subject of a finite clause, but here anyone or anything and hurt you are not subject and predicate; they are respectively direct object and bare infinitival catenative complement of the verb let .
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CliveIn your example, 'hurt' is the base form of the verb.Yes, but that doesn’t explain why it shouldn’t be hurts, does it?
AnonymousIt shouldn't be hurts, not hurt? I mean, hurt goes with anyone.Yes, you are correct. But here the main verb is [let]. In construction like this, the second verb [hurt] doesn't take the third-person "s".
AnonymousI will never let anyone or anything hurt you''."Let" is the main verb.