My best guess for the multipliers that convert a Pillai approximation to an actual number for 10^(10^n), n=3..9, are approximately 2.63659, 2.16878, 1.90743, 1.74774, 1.63974, 1.56111, 1.50095. If this ends up with 1 at infinity, one could possibly curve-fit these numbers into something moderately useful.