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A notion of (average) fractal Euler number for subsets of ℝ  d    with infinite singular complexes is introduced by means of rescaled Euler numbers of infinitesimal  ε  ‐neighbourhoods. For certain classes of self‐similar sets we calculate the associated Euler exponent and the (average) fractal Euler number with the help of the Renewal theorem. Examples like the Sierpinski gasket or carpet are provided. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)