Mrs. Cumbee's Botany Lesson
I found out what the mystery plant is (isn't it wonderful what a little Googling will do!). It's an Agave, which is in the amaryllis family. It is related to aloe vera, which is why it looks so much like one!

The agave can be used in many ways: different varieties provide soap, food (in the grocery stores here, I have seen blades of agave for sale - I just never associated them with this "tree"!), and fiber (the popular sisal comes from agave). The flower stalk produces a juice that, when distilled, is called tequila.

The century plant, as it is also called, has a long lifetime, though not 100 years as its name implies. Plants take from 10 to 50 years to mature and it's at this point that the bloom stalk begins to grow, and does so at a rate of up to two inches a day! After blooming, the plant dies, but new plants will grow up from the fruit it dropped.

My husband, who took the picture below, says he's pretty sure the plant is taller than I estimated. He was only able to get about 50' away from it since it's on private property, but he figured the leafy part of it was at least 4' high, which would make the whole thing 16' or more.