I like this question because it really works your intuition.
The basics: x^y means "grow at x, for y units of time". I see "2^3" as "grow at 2x for 3 units of time".
Having a base of i means your "growth" is a rotation at 90 degrees, no scaling. So i^(1/2) means a 45 degree rotation, i^3 means a 270 rotation, etc.
Raising this to the i power (or 1/i power, which is -i) means the growth that was originally purely rotational is now rotated. So instead of growing at i, you are growing at (i * 1/i = 1). So, we should expect a positive real number, greater than 1 since our growth is positive.
How long do we actually grow for? Well, the base of "i" is really e^(i* pi/2), which means "Start at 1.0 and rotate continuously for pi/2 seconds". We've now modified this to "Start at 1.0 and grow at 1.0 for pi/2 seconds", which is e^(pi/2).
So the answer is i^(1/i) = e^(pi/2) ~ 4.8
It's a bit tough with text-only, read the above article for more diagrams.