Potato Compatible Plants
Potato Plant Companions
Solanum tuberosum Full Sun Soil pH: 4.5 - 6.0
Companion Planting is the growing together of all those elements and beings that encourage life and growth; the creation of a microcosm that includes vegetables, fruits, trees, bushes, wheat, flowers, weeds, birds, soil, microorganisms, water, nutrients, insects, toads, spiders, and chickens." - John Jeavons, How to Grow More Vegetables Than You Ever Thought Possible
Good candidates to companion plant with potatoes include Onions, garlic and Alliums, Horseradish, beans, corn, cabbage and members of the cabbage family, dead nettle, flax, peas, marigold, comfrey, Sweet alyssum.
Bad candidates to companion plant with potatoes include Apple, Celery, Cherry, Cucumber, Pumpkin, Raspberry, Rosemary, Sunflower, Tomato, eggplant, pepper, radishes , turnip, asparagus, cucumber, kohlrabi, pumpkin - squash family, rutabaga.
Planting Onions with Potatoes
Intercropping of Alliums, in particular Onions and Garlic with Potatoes has been demonstrated to have substantial impact on pest populations. Aphids and other sap sucking green flies and their maggot offspring, leafhoppers and bean beetles were repelled from the potato crops. 
Planting Beans with Potatoes.
Inter-planting Potatoes with beans has been proven to increase yields for both. This does not apply to legumes grown below the soil surface, only beans grown above it. 
In addition to their increasing each others yield there also appears to be a symbiosis in deterring each others pests.
"Bush beans planted with potatoes protect them against the Colorado potato beetle. In return, the potatoes protect the bush beans from the Mexican bean beetle. It is considered best to plant the beans and potatoes in alternate rows." 
Excluded from this scenario are snap beans, although they enhance soil quality and so forth, they are susceptible to to the same soil borne diseases as Potatoes. Potatoes and related crops will readily transmit this to the snap beans and vice versa.
Peas and Potatoes Grown Together
Peas can be successfully grown with early Potatoes. Peas repel the Colorado potato beetle. However - Avoid planting them with or near late season potatoes, peas are carriers of Fusarium Crown and Root Rot later in the season.
Planting Horseradish with Potatoes
Horseradish can be planted at the corners of potato patches and yes, it will deter Potato Bugs / Beetles, Blister Beetles and a few other pests. Horseradish also increases the disease resistance of potato plants as it exhibits mild antibacterial and anti-fungal properties.
However, and that's a big "however" - It is not something I would heartily recommend. Horseradish is a very dominating plant with large root systems that readily spread. The smallest fragment of a horse radish root left in the ground can and frequently does produce another plant in subsequent seasons - it becomes an invasive weed - with strong expanding roots. It would be very easy , when digging for potatoes to break of minuscule fragments of the root and contribute to its invasive properties.
Planting Corn and Potatoes
Corn can be successfully planted with Early season potatoes. There are drawbacks - both plants are heavy feeders and neither contributes much in the area of fertilizer residue for the other, so you would need to add fertilizer as they will be very competitive, even if at differing soil levels.
There is a benefit to the potatoes, which are a cool weather crop. A little sun is good, but too much leads to elevated internal leaf temperatures that will actually impede photosynthesis.
Two rows of potatoes to every row of corn will shade the potatoes adequately, not excessively, and prevent overheating.
Cabbage and Brassicas grown with Potatoes
Cabbage can be grown with potatoes purely for the purposes of space management and hedged investment .There are no documented attributes of either plant that would suggest much of a symbiosis other than conjecture and old wives tales. There is no reason however that the two can't be grown together.
Avoid kohlrabi ? Maybe - maybe not - somebody somewhere once said they shouldn't be planted together, but nobody anywhere can say why. So just to be on the safe side ....
Avoid Fennel which is allelopathic to most garden plants. It exudes compounds designed by nature to eliminate competing plants from its immediate area. Dill is one of the few garden crops that can be grown with it.
Companion Planting Potato for Pest and Disease Suppression
Please keep in mind that many inter-actions are also percentage based. If an insect that attacks a certain garden plant is repelled by another type of plant - that doesn't mean that absolutely no insects of the genus in question will go near the plants in question - it simply means a much lower percentage than normally would have attacked the plant will show up.
Colorado Potato Beetle is perhaps the most notorious enemy of potato farmers and gardeners - cilantro, flax, dead nettle, tansy, nasturtium, basil Peas are all deterrents. 
Petunias properties to suppress the Colorado Potato Beetle are very questionable.
Marigolds are helpful in pest suppression in any garden be it veggie or ornamental, however they need to be planted in the same plots for multiple seasons to be effective.
Sweet alyssum can be used against aphids by attracting hover flies whose larva devour aphids. They also draw in pollinators and will reseed themselves without being highly invasive. They make an attractive ground cover.
Raspberry attract potato phytophthora blight. Raspberries Should also not follow potatoes, or related crops in the rotation. Potatoes are susceptible to verticillium wilt, to which most raspberries are highly susceptible.
Cucumbers attract potato phytophthora blight
Squash also attracts potato phytophthora blight. Pumpkins, basically a type of squash and potatoes have an inhibiting effect on each other. 
Comfrey leaves are recommended to prevent scab - this is unproven
Asparagus Contrary to popular belief there is no real reason to avoid planting Potatoes near Asparagus. Like tomatoes, potatoes produce solanine which repels asparagus pests. Other than competition for nutrients, which isn't a factor so long as you don't plant them right on top of each other - there is no reason Potatoes and Asparagus can't be planted in proximity to each other. Related Article - Planting Potatoes and Asparagus - Martha Stewart - Martha Stewart
Crop Specific Companion Planting Guides
Onions Asparagus Garlic Beets Carrots Leeks
Eggplant Tomatoes Cucumbers Gooseberry Horseradish Lettuce
Footnotes and References
1. - Annals of applied Biology - The influence of intercropping with Allium on some insect populations in potato
2.MICROENVIRONMENTAL EFFECTS ON POTATO AND BEAN YIELDS GROWN UNDER INTERCROPPING SYSTEM
3. Yield‐density relationship for potato (Solarium tuberosum) andcommon bean (Phaseolus vulgaris) inintercropping
4. Carrots Love Tomatoes: Secrets of Companion Planting for Successful Gardening- L.Riotte
5. scerlotina [aka cottony rot, watery soft rot, stem rot, drop, crown rot and blossom blight]
6. Effects of planting methods, planting dates and intercropping systems on sweet potato and okra yields
7. COMPANION PLANTING: CON; They Couldn't Care Less - NY Times March 1983