There are several foods popular in St. Louis and not available in many other places. Here are my thoughts on some of them.
Things I grew up eating regularly and like:
Gooey Butter Cake - A super-sweet, gooey coffee cake made of yellow cake, eggs, sugar, and butter, and topped with powdered sugar. Yummy.
Toasted Ravioli - Deep fried ravioli, typically with seasoned beef filling. It is delicious.
Provel cheese - A processed cheese product. It's an acquired taste, and one which is apparently difficult to acquire as an adult--most people who didn't grow up in St. Louis hate it for its flavor, its texture, or both. I love it. It's most often found on St. Louis-style pizza, but it can also be found in salads or melted on almost anything.
Ted Drewes concretes - Frozen custard (basically ice cream made with eggs) mixed with candy, fruit, and/or nuts. They're sort of like extremely thick milkshakes. I grew up eating them in the Ted Drewes parking lot on hot summer nights, and they are wonderful. Some say the concrete was the inspiration for the Dairy Queen Blizzard (the Blizzard was apparently invented by a St. Louis DQ owner).
St. Louis-style pizza - A super-thin crust topped with sweet tomato sauce and provel cheese. It inspires strong feelings in natives and visitors alike--I think it's great, but I've heard others compare it to "Velveeta and ketchup on a cracker." Imo's is the most popular purveyor.
Things I have not eaten (and may never eat):
Slinger - Eggs, hash browns, and a hamburger patty, covered in chili, onions, and cheese. Sounds disgusting.
St. Paul sandwich - Found at some St. Louis Chinese restaurants. Egg foo yung on Wonder Bread with lettuce, tomato and mayo. Sounds disgusting.
Fried brain sandwich- Calves' brains on white bread. It became popular when St. Louis had a lot of stockyards, but it's not very popular anymore, in part because of mad cow disease. Sounds disgusting.