Circles On Circles

New puzzle time? I guess it must be, its certainly not new solution time. Though I think I still have left that cats and dogs game hanging, one day I'll have to revisit that. Anyway, I found this puzzle on the xkcd forums, but its initially from the cofoundry:

There are 16 overlapping circles in a plane with the following proerties:
Circle 1 has radius 1
The diameter of circle 2 is a chord of circle 1
The diameter of circle 3 is a chord of circle 2
...
The diameter of circle 16 is a chord of circle 15
What is the maximum distance from a point in circle 16 to the center of circle 1?

By "overlapping circles" I really mean that circle N can share some interior points with circle N+1, it just helps to say it that way to imagine the setup. If you have forgotten your grade school geometry you may want to look up what a chord of a circle is on wikipedia or something.