The time-resolved images and accelerating propagation speeds of spark-ignited, expanding spherical flames exhibiting both flame-front hydrodynamic and diffusional-thermal cellular instabilities were experimentally acquired in a constant and high pressure environment. From these data the acceleration exponent α, defined through R(t)=C+Atα where A and C are constants, was determined for the near-equidiffusive flames of ethylene and acetylene and non-equidiffusive flames of propane and hydrogen, recognizing that the former is subjected only to hydrodynamic instability while the latter to the diffusional-thermal instability as well. Results show that the acceleration exponent seems to be bounded by the value of 1.34, which is approached for fast-propagating flames of small thicknesses, regardless of the nature of the cells. The characteristic cell sizes were also measured and were found to agree well, for the hydrodynamic cells, with the linear stability theory of Bechtold and Matalon. The possible attainment of a self-similar mode of propagation is suggested.