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Abstract: We consider the percolation model with nucleation and simultaneous growth of multiple finite clusters, taking the initial seed concentration ρ as a tunable parameter. Growing objects expand with constant speed, filling the nodes of the triangular lattice according to rules that control their shape. As growing objects of predefined shape, we consider needle-like objects and “wrapping” objects whose size is gradually increased by wrapping the walks in several different ways, making triangles, rhombuses, and hexagons. Growing random walk chains are also analyzed as an example of objects whose shape is formed randomly during the growth. We compare the percolation properties and jamming densities of the systems of various growing shapes for a wide range of initial seed densities ρ< 0.5. To gain a basic insight into the structure of the jammed states, we consider the size distribution of deposited growing objects. The presence of the most numerous and the largest growing objects is recorded for the system in the jamming state. Our results suggest that at sufficiently low seed densities ρ, the way of the object growth has a substantial influence on the percolation threshold. This influence weakens with increasing ρ and ceases near the value of the site percolation threshold for monomers on the triangular lattice, ρp∗=0.5. Graphical Abstract: [Figure not available: see fulltext.]. © 2022, The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature.