How high is the risk of fire for using a 75 watt bulb in a light rated for maximum 60 watts?

It's more than zero.  The rating is based upon testing and conformance with the corresponding standards. Part of that dictates the label and instructions FOR MINIMAL SAFETY.  Exceeding the rated lamp heat/wattage may only result in faster deterioration of the socket or other parts of the lamp, or may result in a catastrophic fire when the lamp is left switched on for fourteen days in a row.

Your insurance company then investigates, finds out you violated the fire code by failing to follow the manufacturer's instructions, and you're left paying for the damage and injuries out of your own pocket.

Not sure why anyone answers about LED bulbs or other work-arounds (which are certainly safer) since CLEARLY you were specifically asking only about a 75 watt incandescent bulb.

Very low.  If the socket is porcelain then the risk is essentially nonexistent.

If you want more light, change to an LED bulb.  It will give comparable light while using only a fraction of the energy and generating only a fraction of the heat.  Heat is what that wattage rating is trying to control and with an LED lamp the problem is a non-issue.

It could turn brown and get crispy from the heat, that's about it.

Fairly low.  More likely to damage the shade or socket.  Get a 75 watt equivalent LED. 

The fire risk is low--or nonexistent. The bulb will simply burn out faster. It won't catch fire. You'll just keep replacing bulbs. 

too high to fool with.  if you change to LED bulb of similar light output [or even CFL], the excess heat generated is much less and it won't be a problem.  -- grampa

Buy a low wattage LED bulb that produces the same amount of Lumens.

Probably not that high, but it will depend on what kind of lamp and how long you leave it on.

Do not roll the dice.  Put a bulb 60 watts or lower.

Very very very unlikely  ... I have done it many times as I am sure many others have.

If you are still uneasy about it ....Purchase a LED bulb.... there is very little heat generated with those bulbs.