What problem would happen if the ground wire on an outlet was hooked to the netural white wire?

I believe it could put power on the ground circuit. Not good.

Even though they are joined at the main breaker, you don't get power on the ground circuit. Something about the path of least resistance as I understand it. I'm sure someone with a better electrical engineering back ground can explain it much better than I.

If you ground your neutral you will have likely added all of your plumbing and electric conduit to your electrical circuit. Do not do this. What issue were you trying to fix, maybe we can help solve your problem without creating a new dangerous problem.

As bad as that is, only recently have they quit hooking up dryers and ranges that way. On 120 volt circuits, code has long forbid it with good cause. Ground and neutral have different jobs. There is enough safety built into most stuff that it usually takes a couple of things to go wrong before anybody gets hurt or the house burns down. Connecting the ground and neutral is a very effective first step.

Using an equipment ground as a return path instead of the neutral will allow current to be sent back on the equipment ground system continuously. This potentially can cause a shock hazard for anyone coming in contact with a grounded surface with a lower impedance that of the energized frame of an appliance.

Any loose connection in conduit or cable sheathing will create a high resistance connection that will spark and can develop into a fire.

If you are considering doing this as a solution to a broken neutral, or any other reason,do this at your own risk, and be prepared to be responsible for the damaging effects.