Nonreciprocal shape deformations can drive inertialess cellular swimming, as first explored by Taylor and Lighthill in the 1950s, for the small-amplitude squirming of a planar and a spherical surface, respectively. Lighthill's squirmer, in particular, has been extensively studied for large wave numbers in the context of ciliated microbes. The maximal power efficiency for small-amplitude planar squirming motility is well characterized and degenerate, with nonunique optimal swimming strokes. We explicitly show that this degeneracy is retained at high wave numbers for the small-amplitude spherical squirmer such as a ciliated microbe but is broken for low wave numbers. Hence further complexity emerges in parameter regimes outside that of ciliate swimming even at small amplitudes. Large-amplitude squirming also characterizes more recent observations of large-amplitude/low-wave-number membrane deformations driving the motility of Euglena, neutrophils, and Dictyostelium discoideum. Thus boundary element numerical methods are used to explore swimming with increased deformation amplitudes, especially in the context of power efficiency and swimming performance. As radial squirming amplitudes are increased, small-amplitude linearized theories can be unreliable even for nominally low deformation amplitudes. Furthermore, even for a simple single-mode metachronal wave, a highly motile and efficient large-deformation/small-wave-number swimming modality arises, which can surpass theoretical limitations of purely tangential squirming given a constrained surface deformation velocity.