I've seen articles where people who are ferrying planes over long distances get some kind of waiver to take off way over the maximum gross weight for that plane.
How is that determined?
I'm going to try to give you a straight answer, 'cuz I'm kind of sick of PoAers tearing each others' throats up over every issue that gets posted and I want us to get back to being the "front porch of aviation." It's kind of a fun thought experiment too.
The answer to your question: It depends on the airplane. Every aircraft type's gross weight is where it is for different reasons. Even variants of the same type can have their gross at different places for different reasons, and any STC's a particular airframe has may change it as well.
What the published maximum gross weight figure gives you is a known quantity: You'll be able to determine takeoff (and other) performance by the charts in the POH to a reasonable level of accuracy. Your wings will stay attached. You'll be able to go around with full flaps. If you want to know the entirety of the requirements and parameters, read Part 23 of the FARs. If you're like me, that's pretty interesting stuff.
Is it safe to fly over MGTOW? In practice, no. The particular limitation that sets the MGTOW depends what the limitation of the particular plane you're flying is. There's a reason for that MGTOW, and chances are you won't be privy to it, trade secrets of the manufacturer y'know.
Can you do it, and get away with it? Maybe... But how many times is a complete unknown. If you're flying an old, tired airframe that's routinely been flown overweight before (which you won't know), and you load it to 5% over max gross, you may hit a sudden bit of turbulence and find you no longer have a wing attached the very first time you try it. There's a reason the FAA requires a special flight permit to fly over gross, and it won't just be granted automatically. They'll certainly not allow any passengers on the overweight flight, and generally they're going to require a commercial pilot as well.
An example of how unknown these things can be: In the 1960's, a Piper test pilot named Max Conrad flew a Piper Comanche 250 from Casablanca, Morocco to Los Angeles, a distance record for that class of plane which I'm pretty sure still stands. When he took off, the Comanche was loaded over DOUBLE its normal gross weight, and required something like 8,000 feet to get airborne. Tough airplane, right? Shouldn't be a problem to fly one over gross, right? Well, there have also been two structural failures of Comanches in the last 5 years. And Max Conrad's record-setting plane is in a museum now, not flying.
Now, we've all probably unwittingly flown up to 1% over gross - After all, there's probably a lot of dirt and extra wiring and "stuff" unaccounted for in the empty weight of the plane since it may not have been weighed since it was new. And we've probably had pax that lied about their weight, and you're never going to accurately fuel a plane to a less than 1-gallon tolerance so maybe there's an extra six pounds there... Add it all up, and you might be 25 pounds over gross without knowing it.
So does that make it OK to do it on purpose? Well, let's first look at the problem structurally. If you purposely overload 1% over, you still haven't eliminated any of those factors that could cause you to inadvertently be 1% over, and now you're 2% over, and where is the line? Like others have said, the 50% number is the design ultimate load limit. That's the point at which the airplane WILL break if you go past it. At some point significantly less than that - I'd guess it's probably in the range of 20-30% - There will be a *permanent* deformation and weakening of components on the aircraft. If you go 20% over on a regular basis, it won't be long before the plane breaks... Maybe only a couple of flights.
Also, things like Vno and Va are based on a specific upward gust speed. It's quite possible to get into turbulence worse than that, where even flying just barely in the green arc will cause an over-design load. That's probably OK when you're flying at or below gross, because the transient overload gets you only into the range of elastic deformation. But if you've already used up that margin, you may cause damage. "But I'll only do it on smooth days," you say. Well, it's not uncommon to have a day that's smooth as glass and suddenly get a big WHOMP of a bump out of nowhere. It could be the remnants of the wake turbulence from some airliner that flew above that spot 20 minutes ago, who knows... But I've experienced it more than once. I'd hate to lose my wing over it.
Now, more importantly, let's also look at the aerodynamics and performance of each specific flight. As your weight goes up, most of the low-end V-speeds will go up... But the top-end ones (Vno and Vne) will come down. Takeoff performance will suffer the most - In theory, a 10% difference in weight will equal a 21% loss of performance, but in practice it will be even worse since you'll also need to accelerate to a higher Vy and things like prop efficiency, drag, etc. are not going to remain constant throughout the takeoff roll. Climb performance will also suffer greatly - Your Vy will be higher and you'll have increased drag to compensate for (both parasite and induced) and with the increased weight, your excess horsepower will be lower, so your climb rate will be much lower. Some rough calculations for a late-model 172 indicate that being 10% over gross will reduce climb performance by 40%, and that's at sea level... Your service ceiling would be around 5,500 feet.
These are the types of things that a ferry pilot experienced in overweight operations would have to consider prior to embarking on their journey. I hope that this satisfies your curiosity and shows you why it's a bad idea. Fly safe, please!
Not every weight and balance is accurate, fuel gauges are notoriously inaccurate and almost every woman over the age of 12 will lie about her age and weight, most guys will just lie about their weight. 
