If you want to grow record-breaking big tomatoes, follow the lead of Gordon Graham. He holds the Guinness World Records title for the heaviest tomato—a 7-pound 12-ounce whopper grown back in 1986. This winning tomato came from a 'Delicious' vine, so that's obviously a variety to keep in mind. But there are lots of others that are up to the challenge.
Most huge tomato varieties are indeterminate (plants that grow and fruit all season) slicer or beefsteak types. 'Giant Belgium' produces sweet fruits that average 2 pounds. 'Hillbilly' is an heirloom beefsteak type with pink-and-yellow bicolored fruits that weigh in at 1 to 2 pounds. 'Big Zac' is a hybrid of two heirlooms, and it regularly produces 4-to-6-pound fruits! Then there are the tomatoes that I like to call the "big beefies" ('Beefsteak', 'Italian Giant Beefsteak', 'Watermelon Beefsteak', and 'Beefmaster'), which are all well known for producing heavy fruits. Check out Tomato Growers Supply Company (888-478-7333; www.tomatogrowers.com) and Totally Tomatoes (803-663-0016; www.totallytomato.com) for more ideas.
It's a lot of work to coax your tomatoes into really packing on the pounds. First of all, you'll need to provide the best basic care: full sun, fluffy soil with plenty of organic matter, lots of space between plants, an inch of water per week, and support (stakes or cages). Get the plants into the ground early and protect them with Wall-O-Waters or cloches. Hold off on mulching until the ground warms up and the plants begin to flower. Fertilize every two weeks with a dilute fish emulsion fertilizer (but be careful not to provide too much nitrogen, or you'll get a lot of foliage and few fruits).
Once the plants begin to grow, the real training begins. Allow only one stem to develop, and pluck off suckers (the sprouts that form between branches and the main stem) when they are very young. Remove all but two or three fruits from each plant. It's best to eliminate developing fruits at the top of the vine and leave older fruits at the bottom. Prune off tomatoes that develop farthest from the stem and leave one fruit per cluster. Prevent branches from breaking by supporting the tomatoes with pantyhose slings when they start to get really big. And wish for some good luck. You're going to need it if you want a shot at knocking down that record.