It's well known that the expression i^i takes on an infinite set of values if we understand w^z to mean any number of the form exp (z (ln w + 2 pi i n)) where ln is a branch of the natural logarithm function. Since all values of i^i are real, all values of i^i^i (by which I mean i^(i^i))) are on the unit circle, and in fact they form a countable dense subset of the unit circle. I can't figure out what's going on with i^i^i^i, though. I've posted a Mathematica notebook at http://jamespropp.org/iiii.pdf containing some intriguing images starting on page 2. Each image shows the points in the set of values of i^i^i^i lying in an annulus whose inner and outer radii differ by a factor of 10. Can anyone see what's going on? Also, what happens for taller towers of exponentials? Does the set of values of i^i^...^i become dense once the tower is tall enough? Jim Propp