{Abridged} We show that the surface brightness (SB) profiles of elliptical galaxies can be parametrized using a linear superposition of 2-3 components, described by functions developed in Dhar & Williams as the 2D projections of a 3D Einasto profile. For a sample of 23 ellipticals with -24 < Mv < -15, our multi-component models span a range of up to 10^6 in SB and 10^5 in radius, have a median rms of 0.032 mag arcsec^-2, and are statistically justified at >3{sigma}. Our models indicate that i) the central component is more concentrated than the outer component; and ii) the central component of core galaxies is much more luminous, extended and concentrated than that of cuspy galaxies, with their near exponential central profiles indicating disk-like systems whose existence must be verified spectroscopically. While such central excess components are not necessarily contrary to the notion of a mass deficit in core galaxies, we show that the existence, amount, radial extent and sign of mass deficits disagree substantially in the literature, both for a given galaxy and on an average over a sample. We discuss possible implications and suggest that SMBH binaries are unlikely to be the sole mechanism for producing the large cores. We also deduce conditions under which the 3D light density can be described with a multi-component Einasto model for both cuspy and core galaxies; indicating an universality in the functional form of the 3D density of light in galaxies and dark matter in LCDM N-body haloes. Finally, we show that our result - the outer component of the SB profiles of massive galaxies has 5 < n < 8 - could imply i) a common feature of collisionless systems; and ii) that galaxies with such n for their outer component are dark matter dominated.
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