Quantifying the ill-conditioning of analytic continuation

Analytic continuation is ill-posed, but becomes merely ill-conditioned (though with an infinite condition number) if it is known that the function in question is bounded in a given region of the complex plane.
This classical, seemingly theoretical subject has many connections with numerical practice.  One argument indicates that if one tracks an analytic function from z=1 around a branch point at z=0 and back to z=1 again by a Weierstrass chain of disks, the number of accurate digits is divided by about exp(2 pi e) ~= 26,000,000.