You ask a very interesting question. The other answers here point at some fine examples and reasonable explanations. However, I think they only touch on the largest cause for your observation: the human factor.
Assumption
* Most formulas in physics have integer exponents.
No other answer really challenges this hypothesis. To have an irrefutable proof of that hypothesis, we would need to fully enumerate all formulas in physics and divide them into those with integer exponents and those with non-integer exponents. We, as a collective seeker and store of knowledge, are discovering new physics all the time. Will there ever be a final equation? Will there be an end to what we mortal beings can know? I cannot answer these questions.
I will instead deflect your question with a true statement about myself:
* Most formulas that I know have integer exponents.
Amongst the entire population of people alive today, I have an above average understanding of physics. Amongst registered users of this site, my understanding of physics is probably way below average. On an absolute scale of everything humans know about physics, my understanding is laughably incomplete. Regardless of what the absolute count integer vs. non-integer exponents of physical or natural laws comes out to be, I think we can formulate a sound explanation of your observation thus:
Propositions
* Things that are easy to understand have simple representations.
* Integer exponents are easier for humans to understand than fractional, irrational and complex exponents.
* Formulas which have non-integer exponents will be related to or even derived from formulas which have integer exponents.
* Human learning (individually, which necessarily scales to collective learning) begins with understanding the simple things.
* Attempts to understand the complex without understanding of the basics are less successful.
Therfore
* The first formulas humans discovered and recorded which accurately described the natural world used integer exponents.1
* The first formulas one would learn in studying physics (as humans understand it) will have integer exponents.
And Q.E.D., most formulas an arbitrary human knows will have integer exponents.
1: Inquisitive points out an excellent example of this with the sphere. 855
Applying these propositions to that example:
1. A (2 dimensional) circle is easier to understand than a (3 dimensional) sphere.
2. A (1 dimensional) line is easier to understand than a circle.
3. Beginning understanding with a point, then a line, then a circle and a sphere, the equations for each would build upon the most basic, shared elements of each: the center point and the radius.
Therefore, more people know the formula for a circle than a sphere, because it's simpler, easier to understand, and has been a part of human knowledge for longer.
Now that opens the door to the question I think you're really looking to be answered: why? Or more properly stated: Why do the most basic laws of the physical world involve integer exponents?
That depends, I think, primarily on what you mean by basic. From a human perspective, the most basic laws are the ones we learned the first and understand the best, which puts my argument back in play. From the cosmic perspective of which laws are involved in most of the relationships that make up the universe, upon which the macroscopic view depends, the most basic laws are high energy particle physics and quantum mechanics. (Or so I suppose; I think I've already sufficiently declaimed that I know nothing in this field.)
I don't know what the equations in these fields look like. As Inquisitive demonstrates, you could probably find ways to express some or all of the equations in any field with non-integer exponents. However, if and when we arrive at 'the theory of everything,' I would wager the most popular form for the equations, the form that gets recorded in textbooks and taught in classrooms, will involve as many integer exponents as we can make fit.
A delicious example of this is <PERSON>'s formula, which describes trigonometric functions by means of an imaginary exponent: eix = cosx + isinx. Wikipedia summarizes it by saying "This formula can be interpreted as saying that the function eix is a unit complex number, i.e., traces out the unit circle in the complex plane as x ranges through the real numbers." In other words, while the simple, real circle is described with integer exponents, there is another circle describable with imaginary numbers. Who is to say which is more fundamental? Or which leads to more formulas or a broader description of physics and reality? 855