We study the mass distribution of fractal objects growing by subsequent addition of units to the structure. It is demonstrated that in the limit of very large sizes such structures generally exhibit multifractal properties without defining a singular measure on the fractal support. Using the mass as a measure on the fractal, for a few simple deterministic processes we give the spectrum of corresponding fractal dimensions exactly. According to our results such geometrical multifractals should be

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... very common and consist of parts which have a varying local fractal scaling of mass different from that of the whole object. We demonstrate the variation of multifractal properties as a function of the increasing size, in this way establishing for the calculation of multifractality an approach common in the studies of fractal properties of growing clusters.