Fennel
Fennel roots release substances that reliably stunt most garden vegetables, potatoes included. Grow it in its own corner or a pot well away from the bed, where its umbrella-shaped flowers still earn their keep by feeding hoverflies and tiny parasitic wasps.
Sunflower
Sunflowers are tall, greedy feeders whose wide root systems and afternoon shade often leave potatoes short of water and light just as tubers are bulking. Keep them to the north side of the garden or in a separate bed.
Watermelon
Digging potatoes means turning over soil right where melon vines and their shallow feeder roots are running, and the sprawling vines make it hard to reach the rows. Keep the two crops in separate beds.
Cantaloupe
Old charts keep melons and potatoes apart. What a gardener can verify is practical: both are hungry crops on the same water, and digging potatoes tears through the melons' wide, shallow feeder roots and the vines running over the row.
Pumpkin
Digging potatoes means turning over soil right where pumpkin vines have rooted at their nodes, tearing them loose mid-season. Both are heavy feeders as well, so they compete hard for water and nutrients in the same ground.
Butternut Squash
Both are hungry crops that compete for water and food in the same top foot of soil, and digging potatoes in late summer means forking through the shallow squash roots and running vines. Keep them in separate beds.
Zucchini
Both are heavy feeders with wide root systems, so side by side they compete for water and nitrogen. Digging potatoes also tears through the shallow surface roots that squash relies on in midsummer.
Artichoke
Harvesting potatoes means forking through soil where an artichoke has already spread thick perennial roots and crown buds, and that disturbance sets the plant back. Keep potatoes in the annual beds you turn over each year.
Buckwheat
Buckwheat's canopy closes in about three weeks and can out-muscle young potato plants for light and moisture, and volunteer seedlings get in the way of hilling and digging. Grow it before potatoes, not alongside them.
Sunflower Choke
Both host white mold, and both feed the same soil-dwelling wireworms that tunnel tubers, so the pressure often builds where they share ground. Digging potatoes also chops the sunflower choke's shallow rhizomes and scatters unwanted regrowth.
Jerusalem Artichoke
Both hosts of verticillium wilt, a soil fungus that clogs stem tissue and yellows the lower leaves. Stacking them builds the fungus in one patch of ground, and stray sunchoke tubers left behind will sprout through the potato rows for years.
Goji Berry
Potatoes carry nightshade-family wilts and blights, and digging tubers next to a goji both spreads that soil and tears the shrub's spreading surface roots. Keep at least one bed between them.
Raspberry
Potatoes host the soil fungus that causes wilt in raspberries, and the spores persist for years in the ground. Do not plant canes where potatoes have grown within the last three years, and keep the two beds well separated.
Yacon
Both crops host root-knot nematodes, and growing them together or in sequence can build the population that leaves knobby galls on yacon tubers. Digging potatoes mid-season also churns soil right where yacon crowns are working.
Mashua
Both crops are hosts for late blight, and cool wet autumn weather is exactly when mashua is still in leaf. Growing them side by side lets spores move straight from one canopy to the other; keep them in separate parts of the garden.