Examples of functions that are not entire include the logarithm, the trigonometric function tangent, and its inverse arctan.
From the en.wikipedia.org
Intuitively, if one keeps the base fixed and takes the limit as the apex goes to infinity, one obtains a cylinder, the angle of the side increasing as arctan, in the limit forming a right angle.
From the en.wikipedia.org
More examples
Arc tangent: the inverse function of the tangent; the angle that has a tangent equal to a given number