No, it makes sense. exp(i epsilon) = 1 + i epsilon, to first order, i.e. when epsilon is near 0. You can think of 1 + i epsilon as a tiny rotation, when you multiply by it. exp(i t) = the product of a lot of tiny rotations, just as in real numbers exp(x) = the product of a lot of tiny scalings. The product of rotations equals one rotation by the sum of their (individually tiny) angles. As t increases continuously, exp(i t) rotates around from 1 continuously. The unchanging speed of rotation corresponds to how in e.g. bacterial growth, exp(t), the population gets some constant factor bigger in every equal interval.

(I first got this picture from the Feynman lecture on algebra, http://www.feynmanlectures.caltech.edu/I_22.html, which doesn't seem to ever explain it that way explicitly. It's funny how memory works.)