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We prove that the infinitely iterated wreath product in product action of the groups in $\\mathcal{S}$ is topologically finitely generated, provided that the actions of the groups in $\\mathcal{S}$ are not regular. We prove that our bound has the right asymptotic behaviour. We also deduce that other infinitely iterated mixed wreath products of groups in $\\mathcal{S}$ are finitely generated. 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