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EDIT: (my conclusion and thank you note) I want to thank you all guys for this unexpected intellectual and artistic journey. Hope you had fun and enjoyed it the same as I did.

I would like to generate a circle pack that mimics this: (don't pay attention on numbers, and colors at all, at the moment I am interested in circle positions and radii only)

or this:

I am new in Mathematica, could you give me some guidance? Thnx.

EDIT: The question was strictly for planar case (and remains so), however I see @Jacob Akkerboom in his answer added a solution for 3D generalization (thanks!), and, speaking of that, I just want here to bring to your attention this picture:

EDIT 2: There are some applications of circle packing in irregular shapes, like this: (author Jerome St Claire, handpainted)

... and a font called Dotted: (author Maggie Janssen)

... and some logos:

... garden design:

... infographics:

... and these hypnotic images: (from percolatorapp)