Yes, there are urban planter boxes deep enough for growing potatoes—but you need to choose the right depth. Potatoes are not shallow-rooted container crops. They need room for roots and for tubers to form, and most gardeners get better results when they can “hill” soil or mulch around stems as plants grow. In practice, a planter should be at least 12 inches deep; 15–18 inches is better for a small urban harvest, and 18–24 inches is ideal if space and weight allow. A 10- or 12-inch box can work with early varieties and modest hilling, but yields will be smaller and watering will be less forgiving.
For balconies, patios, and rooftops, the best urban options are fabric grow bags (15–25 gallons), galvanized metal raised beds, modular planter boxes, wooden raised beds, and potato towers. Look for containers with drainage holes, because potatoes rot in waterlogged soil. Fill them with loose, fertile potting mix rather than heavy garden soil, and place them in full sun—about six to eight hours daily. Check weight limits before filling large boxes on balconies or roofs; moist soil is heavy. Choose compact or early potato varieties, plant seed potatoes at the right spacing, and hill with compost or straw as stems emerge. Keep moisture consistent, but never leave the box sitting in water. With a deep, well-drained planter and steady care, urban gardeners can absolutely grow potatoes in containers.