  An important component of Ap\'ery's proof that $\zeta (3)$ is irrational
involves representing $\zeta (3)$ as the limit of the quotient of two rational
solutions to a three-term recurrence. We present various approaches to such
Ap\'ery limits and highlight connections to continued fractions as well as the
famous theorems of Poincar\'e and Perron on difference equations. In the spirit
of Jon Borwein, we advertise an experimental-mathematics approach by first
exploring in detail a simple but instructive motivating example. We conclude
with various open problems.
