The study of hard-particle packings is of fundamental importance in physics, chemistry, cell biology, and discrete geometry. Much of the previous work on hard-particle packings concerns their densest possible arrangements. By contrast, we examine kinetic effects inevitably present in both numerical and experimental packing protocols. Specifically, we determine how changing the compression/shear rate of a two-dimensional packing of noncircular particles causes it to deviate from its densest possible configuration, which is always periodic. The adaptive shrinking cell (ASC) optimization scheme maximizes the packing fraction of a hard-particle packing by first applying random translations and rotations to the particles and then isotropically compressing and shearing the simulation box repeatedly until a possibly jammed state is reached. We use a stochastic implementation of the ASC optimization scheme to mimic different effective time scales by varying the number of particle moves between compressions/shears. We generate dense, effectively jammed, monodisperse, two-dimensional packings of obtuse scalene triangle, rhombus, curved triangle, lens, and "ice cream cone" (a semicircle grafted onto an isosceles triangle) shaped particles, with a wide range of packing fractions and degrees of order. To quantify these kinetic effects, we introduce the kinetic frustration index K, which measures the deviation of a packing from its maximum possible packing fraction. To investigate how kinetics affect short- and long-range ordering in these packings, we compute their spectral densities. (chi) over tilde (V)(k) and characterize their contact networks. We find that kinetic effects are most significant when the particles have greater asphericity, less curvature, and less rotational symmetry. This work may be relevant to the design of laboratory packing protocols.