Infinity and infinitely small numbers pique the curiosity of middle school students. From a very young age, many students are intrigued, interested, and even fascinated by extremely large or extremely small numbers or quantities. This article describes an activity that takes this curiosity about infinity into the domain of adding the numbers of an infinite sequence. Some of the activities that the author presents to groups of mathematically advanced students relate in an informal manner to unlearned concepts or procedures. The author stresses that although these activities can be mathematically challenging to students, they provide them with an intuitive, concrete acquaintance that is needed at the initial stages of understanding a mathematical concept. The author presents five stories in the activity that are based on the three never-ending sums (or series): (1) "The Sum of Halves;" (2) "The Sum of Quarters;" and (3) "The Sum of Thirds." The author also describes some issues that were discussed with fifth graders during the ninety-minute lesson on never-ending sums. (Contains 3 figures.)