According to Google, Lagrange "proved" that the most dense packing of equal circles in the plane is given by the familiar hexagonal packing with density 0.906899968.... Laszlo Fejes Toth gave a rigorous proof of that, and considered the case when the radii sizes can be different. If the radii of the circles are close enough to each other, It turns out you might as well use the hexagonal packing since the maximum density for them is the same. In addition to ideas from Laszlo Fejes Toth, and using work of Thomas Fernique, G. Blind, S. Gortler, M. Pierre, A. Florian, T. Kennedy, myself and others, I conjecture that when the radii are between 0.6585340820... and 1.0, then the maximum density is given by the density of the hexagonal packing. (This interval for the radii cannot be enlarged.)