Agave is a desert plant, and Agave Americana is a particularly large species of agave. The leaves grow to be 6 feet long, and have hooked spines on the edges and an especially sharp spine on the end. After about a decade it blooms and promptly dies. The plant does have natural enemies that can kill it, but they are not commercially available. To kill the plant, either dig it up or to wait for it to bloom.
Start the pruning process by cutting off the sharp spine on the ends of the leaves. This is to protect yourself from getting stabbed when you prune down the plant to the ground, which is the next step.
Dig around and under the base, which is like a rosette, not a woody stump. Cut away at smaller roots to make the larger, central roots free enough for pulling out.
Pull out as much of the underground network as you can, using a jack screw or pulling it free with a rope attached to your car. Use a hatchet on smaller roots to help free the central root structure.
Remove remaining roots as much as possible before filling the hole in. Coat the open ends of any remaining roots with Roundup.
Remove smaller roots over the next several months or years, either as they are unearthed from tilling or appear as new plants through the ground. The roots are persistent; and this is a long-term process.
Very Long-term Strategy
Unearth and remove any offshoots that sprout up from the mother plant's underground rhizomes. These offshoots can become mother plants otherwise.
Continue this containment strategy until the plant blooms, which occurs when it is about 10 years old. It will die shortly after blooming.
Maintain the containment strategy especially once the bloom sprouts. It will rise more than 20 feet into the air. From that height, the seeds may spread widely, so be vigilant to prevent reseeding.