Using Gemini North telescope ultra-deep and high-resolution (sub-kiloparsec) K-band adaptive optics imaging of a sample of four nearby (z similar to 0.15) massive (similar to 10(11) M-circle dot) compact (R < 1.5 kpc) galaxies, we have explored the structural properties of these rare objects with unprecedented detail. Our surface brightness profiles expand over 12 mag in range allowing us to explore the presence of any faint extended envelope on these objects down to stellar mass densities similar to 10(6) M-circle dot kpc(-2) at radial distances of similar to 15 kpc. We find no evidence for any extended faint tail altering the compactness of these galaxies. Our objects are elongated, visually resembling S0 galaxies, and have a central stellar mass density well above the stellar mass densities of objects with similar stellar mass but normal size in the present universe. If these massive compact objects will eventually transform into normal size galaxies, the processes driving this size growth will have to migrate around (2-3) x 10(10) M-circle dot stellar mass from their inner (R < 1.7 kpc) region toward their outskirts. Nearby massive compact galaxies share with high-z compact massive galaxies not only their stellar mass, size, and velocity dispersion but also the shape of their profiles and the mean age of their stellar populations. This makes these singular galaxies unique laboratories to explore the early stages of the formation of massive galaxies.