-What kind of apple trees do you have? Some are resistant to some diseases, therefore you might be able to get away with spraying them less.
-What shape are the trees? How neatly do you keep them trimmed? If they have been neatly trimmed and the branches are spread nicely, there is less chance of some diseases. Along the same lines, do they get much wind? Wind through the branches helps keep fungal diseases under control. How are they sited? A well-trimmed tree on a southern slope that gets oodles of sun and a good breeze will rarely need spraying for anything other than bugs. The wind and the sunlight will take care of most other diseases.
-What USDA zone are you in? I am assuming more northerly, as apples need a lot of chill time. However, some areas that don't get many hard freezes also have a lot more bugs. I live in New England, where the long snowy winters tend to keep insects more manageable.
-Do your neighbors have many fruit trees? If, for example, their trees are harboring some diseases, then your trees might also easily become infected. If there ain't another fruit tree for 20 miles, you're in luck there.
Generally I would recommend spraying once or twice with dormant oil in the late fall after the last harvest, then lightly spritz any sickly-looking trees with a bit of Bordeaux mix in the spring, and putting red ball sticky traps around to catch apple maggots.
Also, how many apples are we talking here? I've seen Japanese apples done fancy, where they pick all but about 2 apples off a branch, and those two apples get HUGE. They wrap the apple in an individual paper bag as it grows to keep off bugs, so it stays perfect. It's time-consuming, but it's an option.
Here's a "Home Tree & Small Fruit Management Guide" from the University of Rhode Island. It makes Rosalind's point regarding disease resistance and suggests some apple varieties.
BTW - Sevin (carbaryl) is not mentioned. I'm not much of a synthetic pesticide expert but I believe Sevin has fallen into disfavor because of it is "acutely toxic to honeybees and can destroy colonies of bees that are foraging in an area where the chemical has been applied" according to Wikipedia.
An idea for aphids & spider mites is Palmolive Green at 3 tablespoons per gallon of water.
Here are links to a bunch of good informational sites about growing apples in the home orchard in general, some of them have good info about pest control:
(anyone in the Toronto area, Siloam Orchards has GREAT apples and sells young trees of a gajillion different varieties, both old heritage and the newer-fangled disease-resistant ones, by preorder. HIGHLY recommend )
My mother in law, who has about I dunno 10-15 apple trees in her back yard, many with branches grafted by my formerly grafting-fiend f-i-l, sprays them with dormant oil and bordeaux mixture a coupla times per season. I don't know the schedule but I know at least one round of spraying is right about now. She also sprays some fungicidal thing late summer, slathers it on, for which reason I only eat her apples heavily peeled and cored ;P
Dunno whether bordeaux mix and dormant oil count as 'chemicals' or not, I would sya they are but may be 'organic' enough that they're not conventionally called chemicals <shrug>
Bordeaux mix this time of year. But you don't want to spray that too often or too generously--the salts will build up in the soil. Only spray Bordeaux if you notice a fungus-looking infection. If the new shoots look healthy, leave 'em be.