We define and study infinite root stacks of fine and saturated logarithmic schemes, a limit version of the root stacks introduced by Niels Borne and the second author in Adv. Math. (231 (2012) 1327–1363). We show in particular that the infinite root stack determines the logarithmic structure and recovers the Kummer‐flat topos of the logarithmic scheme. We also extend the correspondence between parabolic sheaves and quasi‐coherent sheaves on root stacks to this new setting.