I've installed a new kitchen light fixture that has a maximum wattage of 60 watts per socket.

Unfortunately this does not create enough light.

What would happen if I up the bulbs to 100 watts each?

It will probably work. However, it could do anything from decreasing the life of the bulb, to starting an electrical fire. It's pretty easy and cheap to take that light fixture back, and install one that handles higher wattage.

It could over heat and start a fire. More likely just discolor the fixture and maybe the ceiling around it. It would also age the insulation on the wires leading to cracking and falling off. A far better solution would be to find compact fluorescent that produce the light of a 100 watt incandescent or more for less than 60 watts. I have found Home Depot has a good selection at a good price. The incandescent bulb's days are numbered. Another good idea if you can, would be to return the fixture and pick up one of the post Edison halogen or xenon fixtures. You might even come across an LED fixture. All of the above use less electricity and much longer life bulbs.