The problem with this question is that static friction and kinetic friction are not fundamental forces in any way-- they're purely phenomenological names used to explain observed behavior. "Static friction" is a term we use to describe the observed fact that it usually takes more force to set an object into motion than it takes to keep it moving once you've got it started.
So, with that in mind, ask yourself how you could measure the relative sizes of static and kinetic friction. If the coefficient of static friction is greater than the coefficient of kinetic friction, this is an easy thing to do: once you overcome the static friction, the frictional force drops. So, you pull on an object with a force sensor, and measure the maximum force required before it gets moving, then once it's in motion, the frictional force decreases, and you measure how much force you need to apply to maintain a constant velocity.
What would it mean to have kinetic friction be greater than static friction? Well, it would mean that the force required to keep an object in motion would be greater than the force required to start it in motion. Which would require the force to go up at the instant the object started moving. But that doesn't make any sense, experimentally-- what you would see in that case is just that the force would increase up to the level required to keep the object in motion, as if the coefficients of static and kinetic friction were exactly equal.
So, common sense tells us that the coefficient of static friction can never be less than the coefficient of kinetic friction. Having greater kinetic than static friction just doesn't make any sense in terms of the phenomena being described.
(As an aside, the static/kinetic coefficient model is actually pretty lousy. It works as a way to set up problems forcing students to deal with the vector nature of forces, and allows some simple qualitative explanations of observed phenomena, but if you have ever tried to devise a lab doing quantitative measurements of friction, it's a mess.)
In general, smoothing the surface, changing the interaction from sliding to rolling, adding (air) space between the surfaces, and adding lubrication (oil, graphite, teflon, ball bearings, air cushion...) are the most common techniques.
Best Answer
As with many problems in physics, the start of understanding is getting the right kind of metaphor and the right level of abstraction.
What is happening when two materials are in contact?
At the level of rough surfaces, in the range of "many" atoms (many thousands up) the contact consists of little parts of one surface poking down past the up-poking bits of the ther surface. In order to move the two surfaces have to separate slightly so the poky-bits can pass by eachother. Unless a few of them break off, such as you get from abrasion. So it can be thought of as little hooks that can catch on to the other surface. You have to pull these away before the surfaces can start to slide.
At a smaller level, in the range of "few" atoms (some tens or so) there are various low energy bonds that can form. It is not typically forming new molecules. It's just that there are places on the molecules of one substance such that they attract the molecules of the other. Rubber and similar products have very large and complicated molecular structures. Often even very highly diverse molecular structures, almost haphazard. This provides huge opportunity for the molecules of the other side to find places that are slightly lower energy when they are close together. These act as a huge number of much smaller hooks.
Thse effects can be significantly increased if one of the surfaces can deform a bit so as to conform to the other very tightly. This is part of how such things as rubber tires manage to grip the road.
When the two surfaces are sliding these various hooks have less chance to find good purchase.
This indicates when you should expect to find the surfaces have little difference between static and kinetic friction. The surfaces should be rigid to reduce deformation. They should have very little surface roughness. The molecules should not have readily available interaction sites for the other surface to latch onto.
And @Bob D has given us a link. Thanks Bob! There we find polytetrafluoroethylene otherwise known as teflon. It has 0.04 for both static and kinetic. This is also a non-stick coating used in cookware. The basic idea is that it seriously reduces both the mechanical and molecular "hooks" holding one surface to the other. This tends to make friction low for both, while reducing the difference.